A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods

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A Flux Reconstruction Approach to High-Order Schemes
Including Discontinuous Galerkin Methods
H. T. Huynh*
NASA Glenn Research Center, MS 5-11, Cleveland, OH 44135, USA; E-mail: huynh@grc.nasa.gov
[Abstract] We introduce a new approach to high-order accuracy for the numerical
solution of conservation laws by using flux reconstruction. The approach bridges the two
existing approaches of discontinuous Galerkin (DG) and staggered-grid (or spectral
difference), and leads to simplified versions of these two methods. It also results in several
new schemes with favorable properties. Schemes via the current approach are simple and
economical: they solve the conservation laws in differential form and involve only one grid
(not two grids that are staggered). In addition, they are conservative. As a result of the new
approach, the DG scheme can be formulated using the differential instead of integral form.
Another result is a new method, which is similar to the staggered-grid scheme, but more
accurate, economical and stable.
I. Introduction
In the field of Computational Fluid Dynamics, low-order methods are less accurate, but generally are robust and
reliable; as a result, they are routinely employed in practical calculations. High-order (third and above) methods
have the potential of providing accurate solutions at reasonable cost; however, they are more complicated and less
robust. The need to improve and refine high-order methods as well as to develop new ones with more favorable
properties has attracted the interest of many researchers. The current work is a step in this direction.
For the case of one spatial dimension, in a high-order semi-discretization of conservation laws, the solution can
be approximated in each cell (interval) by the data at
K
points called solution points, where
K
is a positive integer.
These
K
pieces of data define a polynomial of degree
1
−
K
, herein called the solution polynomial. This
polynomial interpolates the data within a cell, a feature of the finite-element methods; in contrast, the finite-volume
methods employ interpolation across several cells. The solution polynomials collectively form a function, which can
be and usually is discontinuous across cell interfaces. At each interface, from the two values to its left and right, a
common flux, which is typically the upwind flux, a key feature of the finite-volume methods, is defined. The
common or upwind flux allows the data in adjacent cells to interact and its commonality results in a conservative
scheme.
The upwind flux can be employed in two different approaches. The first approach, introduced by Reed and Hill
for neutron transport equations (1973) and developed for fluid dynamics equations by Cockburn and Shu and others
(Cockburn, Karniadakis, and Shu 2000 and the references therein), is the discontinuous Galerkin or DG approach.
Here, after multiplying the conservation law with a test function, integrating over a cell, and applying an integration
by parts, the upwind fluxes are employed to evaluate the boundary integral. For the two-dimensional case, the
approach can be formulated on triangular as well as quadrilateral meshes. If the mesh is quadrilateral, and the
solution points are chosen to be the tensor products of the Gauss points, the DG schemes become highly economical
and are called the discontinuous spectral element methods (Kopriva, Woodruff, and Hussaini 2000).
The second approach, introduced by Kopriva and Kolias (1996), solves the differential form of the conservation
law as opposed to the integral form, and is referred to as the staggered-grid approach. (Note that this type of
staggered grid is completely independent of the type commonly employed in incompressible flows where the
location for pressure is staggered relative to that of velocity.) Here, the solution points are the
K
Chebyshev-Gauss
points; the flux points, which include the two cell interfaces, are the
1
+
K
Chebyshev-Lobatto points; and these two
sets of points are staggered. In each cell, the solution polynomial is of degree
1
−
K
and is defined by the values at
the solution points. The flux polynomial is of degree
K
and defined by the upwind flux values at the two interfaces
together with the flux values at the
1
−
K
interior flux points. The spatial derivatives of the flux polynomial at the
18th AIAA Computational Fluid Dynamics Conference
25 - 28 June 2007, Miami, FL AIAA 2007-4079
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

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solution points are employed to update the solution. For the case of triangular meshes, this approach is more
involved and results in the spectral difference schemes (Liu, Vinokur, and Wang 2003, 2005).
The exact relationship between the DG and staggered-grid schemes has not been established. The reason,
perhaps, is that the two approaches are very different. The DG approach is highly mathematical and provides
powerful machinery to obtain the solution. However, the approach is not intuitive; e.g., it is not clear how
oscillations are generated near a discontinuity. The staggered-grid approach, on the other hand, is intuitive; e.g., one
can easily see how oscillations are generated near a discontinuity by plotting the flux polynomials. Mathematically,
however, it remains to be shown whether the schemes using staggered grids are stable and/or optimal.
In addition to the above differences, there is also the following trade-off: the DG scheme is more accurate, but
the staggered-grid has a larger CFL limit. (The CFL or Courant, Friedrichs, and Lewy limit is the largest
nondimensional time step allowed by stability condition.) Moreover, for scalar conservation laws, but not for the
Euler equations, the DG scheme is super-accurate: its order of accuracy is
1
2
−
K
(Biswas et al. 1994, Adjerid et al.
2002). A scheme using
K
solution points is said to be super-convergent or super-accurate if its order of accuracy is
higher than K. The staggered-grid scheme, as will be shown below, is not super-accurate. In general, it is not well
understood what makes a scheme super-accurate or whether there is a relation between super-accuracy and CFL
limit.
In this paper, we introduce a new flux reconstruction approach, which is considerably more general than that of
the staggered grid scheme. The approach establishes the relation between the two existing approaches of DG and
staggered-grid and leads to simplified versions of these two methods as well as to several new schemes. Schemes via
the new approach are simple and economical: they solve the conservation laws in differential form and involve only
one grid, namely the solution points. In addition, they are conservative. As a result of the new approach, the DG
scheme can be formulated using the differential instead of integral form. The key ideas in the one-dimensional case
are the following. The flux values at the Ksolution points define a polynomial of degree
1
−
K
. These polynomials
collectively form a function, which is discontinuous across cell interfaces. Approximating the derivative of the flux
by that of this discontinuous function results in bad solutions since such a derivative ignores the neighboring data; in
other words, it involves no interaction with the data in adjacent cells. To account for the interaction (or the jumps at
interfaces), we first construct a continuous flux function, which approximates the discontinuous function in some
sense. At each interface, we require this continuous function to take the upwind flux value; thus, continuity across
interfaces is assured. In each cell, the continuous flux function must be a polynomial of degree Kso that its (spatial)
derivative is of degree
1
−
K
, consistent with the degree of the solution polynomial. By considering the difference of
the continuous and discontinuous flux functions, it suffices to define a correction function g, which approximates the
zero function. Focusing on the correction due to the jump only at the left boundary, factoring out the size of the
jump, and assuming the cell is the interval ]1,1[
−
=
I
, the problem reduces to defining a polynomial gof degree K
such that 1)1( =−g, 0)1( =g, and gapproximates the function 0 in some sense. The condition 1)1( =−gdeals with
the jump at the left interface whereas 0)1( =gleaves the right interface value unchanged. Since two conditions are
set, gis determined by
1
−
K
additional conditions. At each solution point, the derivative of gis then employed to
correct that of the discontinuous flux function. The correction due to the jump at the right boundary follows by
symmetry. Note the following differences compared to the staggered-grid approach: the use of the discontinuous
flux function; the separation of the correction at the left interface from that of the right; the introduction of the
correction function
g
; and the use of only one grid. While the best choice for
g
is likely to be problem dependent,
we present several choices and discuss the trade-offs.
The first choice for
g
results in a simplified version of the DG scheme and can be described as follows. Denote
by m
Pthe space of polynomials of degree mor less on
I
with the integral inner product. If the
1
−
K
additional
conditions are that the projection of
g
onto 2−K
Pis 0 (i.e., gapproximates the function 0 in the least-squares
sense), then gis the right Radau polynomial, and the resulting scheme is the DG method. Alternatively, this scheme
can be obtained by defining the continuous flux function in each cell via the two interface upwind fluxes and the
1
−
K
conditions that its projection onto 2−K
Pis the same as that of the discontinuous flux function. The current
formulation of DG via the correction function gis simple and economical: it uses only derivative formulas at the
solution points and involves no quadratures. In fact, it is as economical as the spectral element scheme (Kopriva et
al. 2000) with the added advantages of conceptual simplicity and coding simplicity and economy regardless of the
choice of solution points.
Kopriva and Kolias’ staggered-grid scheme can also be cast in this approach: the
1
−
K
additional conditions for
g
are that
g
vanishes at the
1
−
K
interior Chebyshev-Lobatto (flux) points; that is, gis defined by interpolation
rather than projection (least squares). The current version of the scheme is, as in the DG case, simpler than the

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original: we deal only with one grid, namely, the solution points, and they can be the Gauss, Lobatto, or equidistant
points. (Note that for our version, the name staggered-grid is no longer appropriate.) Via Fourier analysis, we also
show that the staggered-grid scheme is mildly unstable. This problem can be remedied by one of the new methods:
the scheme (with gdefined via interpolation) using the two cell boundaries and the
1
−
K
Legendre-Gauss points as
flux points is stable. This scheme is accurate to order
2
2
−
K
. Compared to DG, its accuracy is one order lower, but
to its advantage, the CFL limit is roughly 3/2 times as large.
Among new members, the scheme using the
K
Legendre-Lobatto points as solution points, together with a
correction function that lumps the derivative corrections to the boundary, is simplest: the interface values are readily
available without interpolation, and the derivative corrections only take effect at the corresponding boundary and not
at any other solution points. This scheme is also accurate to order
2
2
−
K
. Compared to DG, its accuracy is one
order lower, but to its advantage, the CFL limit is roughly twice as large. It is also simpler than the schemes using
Lobatto points in the literature (e.g., Hesthaven 1997, Kopriva 1994, and Huang, Wang, and Liu 2005).
The next result, obtained via Fourier (Von Neumann) analysis, provides a criterion for super-accuracy. If the
correction function gis perpendicular to m
Pwhere 20
−
≤
≤
Km , then the scheme is accurate to order 1
+
+
mK .
In particular, the DG scheme corresponds to the choice of Radau polynomial for gand, since this polynomial is
orthogonal to 2−K
P, the scheme is accurate to order
1
2
−
K
, a fact consistent with its super-accuracy.
Concerning stability, Fourier analysis for values of
K
up to 10 leads to the following observation: schemes that
are accurate to order
1
2
−
K
and
2
2
−
K
are stable; schemes accurate to order 32
−
Kor lower usually encounter
instabilities which, for well constructed schemes, are mild. For optimal or ‘best of class’ schemes, we have the
following trade-off: the higher the super accuracy, the smaller is the CFL limit. Another observation is that a steeper
correction function tends to result in a scheme with a smaller CFL limit.
We then show that for advection, schemes via flux reconstruction can be cast in the DG framework. The
difference is the space of test functions. For DG, this space is 1−K
P, which is of dimension
K
. For the scheme
defined by the correction function
g
, this space is also of dimension
K
, but consists of polynomials
φ
of degree K
such that
dx
d
/
φ
is perpendicular to
g
. If
g
is the Radau polynomial, the scheme reduces to DG, and this space
reduces to 1−K
P.
Moving on to the case of two dimensions, the new flux reconstruction approach can be extended to quadrilateral
meshes. Fourier analysis on a rectangular mesh reveals that for schemes via flux reconstruction, the order of
accuracy of the scheme is identical to that of the one-dimensional case. The CFL limit, however, is reduced by a
factor of 2/1 , and this reduction takes place due to the case of flow along the diagonal direction. While this
reduction in CFL limit for the piecewise constant scheme is well known, the result for the high-order case is new.
Finally, the extension to the case of a triangular mesh remains a topic of future research since the concept of tensor
product no longer applies.
In the hope of attracting the interest of researchers who are not familiar with these types of high-order schemes,
this paper is written in an essentially self-contained manner. It is organized as follows. Section II deals with
conservation laws, upwind fluxes, and various flux values to the left and right of each interface. Section III
introduces the correction function
g
. Section IV describes various choices of correction functions. Fourier analysis
for a general correction function is provided in Section V. Stability and accuracy analyses of schemes corresponding
to various correction functions are given in Section VI. Section VII includes two proofs: (a) the flux reconstruction
scheme with
g
defined by the Radau polynomial is identical to the DG scheme and (b) for advection, flux
reconstruction schemes with arbitrary
g
can be cast in the DG framework. Section VIII discusses 2D extensions.
Section IX contains numerical tests. Conclusions and discussions can be found in Section X. Finally, reviews of the
Legendre, Radau, and Lobatto polynomials, and corresponding quadrature points, as well as derivations of new
polynomials and properties needed are provided in the Appendix.
II. Conservation Laws and Upwind Fluxes
Consider the conservation law 0=+ xt fu (2.1)
with initial condition )()0,( init xuxu =where )(
init xu is periodic or of compact support so that boundary conditions
are trivial. The flux fis assumed to depend on u. Denoting the signal speed dudf / by )(ua , the above can be cast
in nonconservation form:

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0=+ xt auu .(2.2)
Let the domain of calculation be divided into (possibly nonuniform) cells or elements j
E,...,2,1,0,1...
−
=
j
On
each cell j
E, let the solution be approximated by Kpieces of data kj
u,,
K
k
...,,1
=
at locations kj
x,, which are
called solution points. The Ksolution points are typically the Gauss or Lobatto points, but equidistant points can
also be employed. In fact, the Fourier stability and accuracy results below are independent of the type of points
chosen. We assume that the same type of points is employed for all cells. Note that if the Lobatto points are selected,
then since they include the two cell interfaces, each cell interface 2/1+j
xcorresponds to two values of u, namely,
Kj
u,and 1,1+j
u, and these left and right interface values are readily available for the upwind flux calculation.
At each solution point, the solution )(
,tu kj depends on
t
. For simplicity, )(
,n
kj tu is abbreviated to kj
u,. At
time level n, suppose the data kj
u,are known for all jand k. We wish to calculate dttud kj /)(
,at time n
tt =,
which is abbreviated to dtud kj /
,. In other words, we wish to evaluate x
fat the solution points kj
x,in terms of
the data. Then, we march in time by, say, a Runge-Kutta method.
As is standard in finite element methods (Hughes 2000), instead of dealing with the cell j
E(global element), it
is often more convenient to deal with the interval ]1,1[
−
=
I
(local element). Denote the center of j
Eby j
xand its
width, j
h. With
ξ
varying on
I
and
x
, on j
E, the linear function that maps
I
onto j
Eand its inverse are
2/)( jj hxx
ξξ
+= and jj hxxx /)(2)( −=
ξ
.(2.3a, b)
The solution points on
I
(in the local description) are denoted by k
ξ
,Kk ...,,1=. They relate to the solution points
on j
E(in the global description) by
2/)(
,jkjkkj hxxx
ξξ
+== .(2.4)
A function )(xrjon j
Ein the global description results in a function in the local description denoted by )(
ξ
j
rfor
simplicity of notation: )())(()( xrxrr jjj ==
ξξ
.(2.5)
To avoid confusion, the function j
ris usually referred to as )(xrjor )(
ξ
j
r. The derivatives in the global and local
descriptions are related by the chain rule,
ξ
ξ
d
dr
hdx
xdr j
j
j)(
2
)( =. (2.6)
Returning to solving the conservation law (2.1), the first task is to approximate uin the cell j
Eby a polynomial
of degree
1
−
K
, which interpolates the Kpieces of data kj
u,,
K
k
...,,1
=
. To this end, for each k, let kj,
φ
be the
Lagrange polynomial on j
Eof degree
1
−
K
that takes value 1 at kj
x,and 0 at all other solution points. These kj,
φ
are called basis (or cardinal or shape) functions. In global and local descriptions respectively, they are
∏∏ ≠=≠= −
−
=
−
−
=K
kll lk
l
k
K
kll ljkj
lj
kj xx
xx
x
,1,1 ,,
,
,)(and)(
ξξ
ξξ
ξφφ
.(2.7a, b)
On j
E, let )(xu jbe the polynomial of degree
1
−
K
interpolating kj
u,,
K
k
...,,1
=
;)(xu jis called a solution
polynomial. In global and local coordinates, respectively,
 ==
== K
k
kkjj
K
k
kjkjj uuxuxu
1
,
1
,, )()(and)()(
ξφξφ
.(2.8a, b)
Basis functions for the case
4
=
K
using Lobatto points and the solution polynomials are shown in Fig. 2.1.
Next, we define the discontinuous flux function. From the data kj
u,at the solution points, we can easily
evaluate the corresponding fluxes: )( ,, kjkj uff =.(2.9)
Similar to (2.8), we define the flux polynomial )(xf jto be the polynomial of degree
1
−
K
interpolating the values
kj
f,at the solution points,
K
k
...,,1
=
. Again in global and local coordinates, respectively,

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 ==
== K
k
kkjj
K
k
kjkjj ffxfxf
1
,
1
,, )()(and)()(
ξφξφ
.(2.10a, b)
Note that both the flux and solution polynomials )(xf jand )(xu jare of degree
1
−
K
. The flux polynomials
)(xf j,...,2,1,0,1...
−
=
j
form a function, which is generally discontinuous across cell interfaces and is called the
discontinuous flux function.
- 1
- 0 . 5
0
0 . 5
1
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1Lobatto
(a) (b)
Figure 2.1. (a) Cubic basis functions.Here, the solution points are the Lobatto points on ]1,1[
−
=
I
for
4
=
K
.
Note that at each interior Lobatto point, the corresponding basis function reaches a local maximum. (b) Solution
polynomials. Cubics )(
1xu j−and )(xu j.
If we ignore the jumps and calculate and employ the derivative of the discontinuous flux function, we obtain
erroneous results. In other words, for the conservation law (2.1), if we employ xj
f)( to evaluate x
f, we get poor
solutions because such a derivative does not include the interaction of the data between adjacent cells. With such a
derivative, the solution never feels the effect of the boundary conditions. In fact, it can be worse as shown in the
following example. If
1
=
K
, i.e., the case of a constant function in each cell, then by ignoring the jumps, we have
0)( 1, =
jx
ffor all cells; as a result, the solution does not change with time, which is incorrect. To account for the
interaction, i.e. the jump, we will construct a continuous flux function, which approximates the discontinuous
function in some sense, and then calculate its derivative. The continuous flux function will be obtained by adding a
correction to the discontinuous one. As a result, we still need the derivative of the discontinuous function, which is
given below. At each solution point k
ξ
of the local coordinate, Kk ≤≤1,

=
=K
l
ljlkkj fdf
1
,,
)(
ξ
.(2.11)
That is, kj
f,
)(
ξ
is given by the dot product of the ( th−krow) vector K
llk
d1
}{ =with the (column) vector K
llj
f1, }{ =.
The coefficients lk
dform a
K
K
×
matrix denoted by Dand is called the derivative matrix (for the derivative
operator on
I
). This matrix can easily be calculated via a software package such as Mathematica or Maple using
(2.10). By (2.6) or the chain rule, for the discontinuous flux function, in the global coordinate,

=
=K
l
ljlk
j
kjx fd
h
f
1
,, 2
)( .(2.12)
For each
j
, denote the flux and the flux derivative vectors, which are column vectors of
K
entries, respectively by
K
kkjxjx
K
kkjj ff 1,1, }){()(and}{ == == ff . (2.13a,b)
Then, the vector of derivatives of the discontinuous flux function at the solution points is given by
j
j
jx hfDf 2
)( =.(2.14)
Another way to evaluate }){( ,kjx
fis by the chain rule: with K
kkjx
u1, }){( =given by a formula similar to the above,
kjxkjkjxkjukjx uuauff ,,,,, ))(()()()( == .(2.15)
)(xu j
)(
1xu j−
ξ
η
x
u

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6
The continuous flux function to be defined is required to take the upwind flux values at the interfaces. In
addition to the upwind flux values, we also need various left and right interface values discussed below. At the
interface 2/1+j
x, let the subscripts Land Rdenote the values just to its left and right, respectively. That is, let L
u
and R
ube the values at 2/1+j
xtaken on by the solution polynomials )(xu jand )(
1xu j+respectively,
)(and)( 2/11,2/12/1,2/1 +++++ ==== jjRjRjjLjL xuuuxuuu .(2.16a, b)
Similarly, let L
fand R
fbe the values at 2/1+j
xtaken on by the flux polynomials )(xf jand )(
1xf j+respectively,
)(and)( 2/11,2/12/1,2/1 +++++ ==== jjRjRjjLjL xfffxfff .(2.17a, b)
In the local coordinate
ξ
on ]1,1[
−
=
I
, we have
)1(and)1( 1,2/1,2/1 −== +++ jRjjLj ffff .(2.17c, d)
In general, RL uu ≠, and RL ff ≠. Since L
uis defined by )(xu j,L
fby )(xf j,R
uby )(
1xu j+, and R
fby
)(
1xf j+, again, in general,
)(and)( RRLL uffuff ≠≠ .(2.18a, b)
We must exercise caution in the choice of using )( L
uf or L
f. At each interface, as will be shown, we use )( L
uf
and )( R
uf for the definition of the upwind flux. For the jump in flux value employed with the correction function,
however, we use L
ff −
upw and R
ff −
upw as will be shown in (3.8) to assure that the scheme is conservative.
We can now derive one of the most popular upwind fluxes, namely, Roe’s flux (Roe 1986). At the interface
2/1+j
x, with L
uand R
ugiven by (2.16), let u
~
be the value that satisfies the mean value theorem:



=
≠
−
−
=otherwise.)()(
,if)())()((
)
~
(
RL
RLLRLR
uaua
uuuuufuf
ua (2.19)
(For the Burgers equation, 2/)( 2
uuf =; consequently, 2/)(
~
RL uuu += .) The upwind flux is determined by the
wind direction, i.e., the sign of )
~
(ua :



≥
== +otherwise.)(
,0)
~
(if)(
upw,2/1upw R
L
juf
uauf
ff (2.20)
In the above expression, we split the flux based on the sign of )
~
(ua . If 0)
~
(=ua , then by (2.19), )()( RL ufuf =.
As a result, it does not matter which of )( L
uf and )( R
uf is selected. By using (2.19) and unifying the cases of
0)
~
(≥ua and 0)
~
(<ua , the above upwind flux can be written with no conditional statement:
)(|)
~
(|])()([ 2
1
2
1
upw LRRL uuuaufuff −−+= .(2.21)
This expression is called flux difference splitting.
Note that if we split the flux, then when 0)
~
(=ua , it is still well defined. However, if we split the variable
u
,
then when 0)
~
(=ua , it is not clear which of L
uor R
ushould be selected. Therefore, we do not split
u
.
For the Euler equations, u
~
in (2.19) turns into Roe's tilde state, and the upwind flux via (2.20) or (2.21) becomes
Roe's flux. Note in passing that for Roe’s scheme, the condition UAF ∆=∆
~
(similar to (2.19)) between the flux
and conservative variables is well known; however, the condition VMU ∆=∆
~
between the conservative and
primitive variables, which simplifies the derivation, is not (see Huynh 1995). Also note that the upwind flux given
by (2.21) generally requires an entropy correction, which is beyond the scope of this paper.
III. Flux Correction Functions
The upwind flux can be employed in (at least) two different approaches. The first approach deals with the
integral form. Here, after multiplying the differential conservation law with a test function, integrating over a cell,
and applying integration by parts, the upwind fluxes are employed to evaluate the boundary integral. This approach
results in the discontinuous Galerkin or DG methods (see Cockburn, Karniadakis, and Shu 2000 and the references
therein) and the discontinuous spectral element methods (Kopriva, Woodruff, and Hussaini 2000). The second
approach solves the differential form of the conservation law. Here, to account for the data interaction between
adjacent cells, we construct a continuous function denoted by )(xF for the flux, which takes the upwind flux values

American Institute of Aeronautics and Astronautics
7
at the cell interfaces. In each cell
j
, )(xF is a polynomial denoted by )(xFj, which approximates the
discontinuous flux function )(xf jdefined in (2.10). The staggered-grid scheme of Kopriva and Kolias (1996) and
spectral difference scheme of Liu, Vinokur, and Wang (2003, 2005) employ this approach and reconstruct the
continuous flux function )(xF by requiring it to interpolate the fluxes at the
1
−
K
interior Chebyshev-Lobatto
points.
Note that the concept of ‘recovery’ presented by Van Leer and Nomura (2005) and Van Leer, Lo, and Van
Raalte (2007) for the diffusion problem and the concept of ‘reconstruction’ discussed in this paper are similar in that
both approaches seek to construct a function that represents the data better than the discontinuous piecewise
polynomial function. The key difference, however, is that the ‘recovery’ spans across adjacent cells whereas the
‘reconstruction’ stays within a cell (but includes the two cell interfaces).
We now introduce a new flux reconstruction approach, which is considerably more general than that of the
staggered grid scheme. The approach leads to several new schemes as well as simplified versions of the DG and
staggered-grid methods. As a result of the new approach, the DG scheme can be formulated using the differential
form, and such a formulation involves no quadratures. For efforts to avoid quadratures, see (Atkins and Shu, 1998)
and (Dumbser and Munz, 2007).
To describe the approach, first, as mentioned above, the continuous flux function )()( xFxF j
=is required to
take the upwind flux values at the two cell interfaces:
upw,2/12/1upw,2/12/1 )(and)( ++−− == jjjjjj fxFfxF .(3.1a, b)
Next, what degree should the polynomial )(xFjbe? To answer this question, in the cell
j
, we allow the
solution polynomial )(xu jto depend on
t
, i.e., for each fixed
t
,),( txu jis a polynomial of degree
1
−
K
. The
conservation law (2.1) can then be approximated by
0))((),( =+
∂
∂
xjj xFtxu
t
.(3.2)
After one (Euler explicit) time step of size
τ
(one stage of the explicit Runge-Kutta time stepping), denoting the
solution polynomial at the new time by )(
new xu j, the above implies
xjjj xFxuxu ))(()()(
new
τ
−= . (3.3)
Here, the right hand side is evaluated at time n
t, and )(xu jis of degree
1
−
K
. For the solution )(
new xu jto also be
of degree
1
−
K
, since
τ
is a constant, the function xj xF ))(( must be a polynomial of degree
1
−
K
in
x
.
Therefore, )(xFjis required to be of degree
K
. It is also required to approximate )(xf j, which is of degree
1
−
K
.
(a)
-1 -0.5 0.5 1
-
0.4
-0.2
0.2
0.4
0.6
0.8
1
(b)
Figure 3.1. (a) Discontinuous and continuous flux functions.Here, the conservation law is the advection
equation 0=+ xt uu . The solution points are the Gauss points with
4
=
K
. The curve interpolating flux values at
the solution points (small dots) is the discontinuous flux function )(xf j, which, in this case, is the same as the
solution polynomial. It is of degree
1
−
K
. At the interface, the upwind flux is the flux from the left cell. The (blue)
curve connecting the big dots is the continuous flux function )(xFj, which is a polynomial of degree
K
. At each
solution point, x
f of the conservation law is approximated by the derivative of )(xFjas shown by the thick (light
blue) line segments. Each of these slopes can be obtained by adding a correction term to the slope of the
discontinuous flux function. (b) A correction function for the left boundary.Here,
g
is the right Radau
polynomial. The thick line segments correspond to the values of the derivative of
g
at the solution points. Note the
alternating signs of the slopes.
)(xf j
)(xFj
f
x
)(
1xf j−upw,2/1−
j
f
LB
gg =
ξ

American Institute of Aeronautics and Astronautics
8
Note that in (3.3), both sides are polynomials of degree
1
−
K
. These polynomials can be described by using
K
solution points of any type. The accuracy and stability results via Fourier analysis, as will be shown in Section 6, are
independent of the type of solution points chosen. In fact, similar to the DG scheme, instead of using the solution
points, we can use the Legendre polynomials as basis functions as well. The derivative of the correction function
g
must then be expressed in terms of the Legendre polynomials. For the 1D case, the Legendre polynomials work
well; for the 2D case, however, the flux reconstruction schemes using the Legendre polynomials are not as simple as
those using solution points.
As discussed above, the continuous flux function )(xF is required to take the upwind flux values at cell
interfaces. In each cell j
E, it is a polynomial of degree
K
denoted by )(xFj, and it approximates the
discontinuous flux function )(xf j. Instead of defining )(xF , we define )()( xfxF jj −, which approximates the
function 0. Switching to the local coordinate
ξ
on ]1,1[
−
=
I
, at the two interfaces
±1
=
ξ
, the function
)()(
ξξ
jj fF −takes the following values which, loosely put, are the jumps:
)1()1()1(and)1()1()1( upw,2/1upw,2/1 jjjjjjjj fffFfffF −=−−−=−−− +− .(3.4a, b)
Therefore, )()(
ξξ
jj fF −is a polynomial of degree
K
with prescribed left and right interface values.
We now separate the prescription of the jump at the left interface from that of the right. Again, using the local
coordinate
ξ
on
I
, for the left interface, let LB
gbe the correction function (LB for ‘left boundary’) defined by
,0)1(,1)1( LBLB ==− gg (3.5a, b)
and LB
gis a polynomial of degree
K
approximating the constant function 0 in some sense. Similarly, let RB
gbe
the correction function for the right boundary:
,1)1(,0)1( RBRB ==− gg (3.6a, b)
and RB
gis also of degree
K
and approximating 0 in some sense. Consider the left interface 2/1−j
x, i.e., 1
−
=
ξ
.
The polynomial
))−−+) −
ξξ
(]1([( LB
upw,2/1 gfff jjj (3.7)
provides a correction at the left interface for )
ξ
(
j
fby changing the flux value at this interface from )−1(
j
fto
upw,2/1−j
f, while leaving the value at the right interface unchanged, namely, )1(
j
f. Next, the polynomial
))−+))−−+)=) +−
ξξξξ
(]1([(]1([(( RB
upw,2/1
LB
upw,2/1 gffgfffF jjjjjj (3.8)
provides corrections to both interfaces: using (3.5) and (3.6), one can easily verify that upw,2/1
)1( −
=− jj fF and
upw,2/1
)1( +
=jj fF . Thus, the above )
ξ
(
j
Fis of degree
K
, takes the upwind flux values at the two interfaces, and
approximates )
ξ
(
j
fin the same sense that LB
gand RB
gapproximate the zero function.
Finally, the derivative of )
ξ
(
j
Fat the solution point k
ξ
is
)
′
)−+)
′
)−−+= +− kjjkjjkjkj gffgfffF
ξξ
ξξ
(]1([(]1([)()( RBupw,2/1LBupw,2/1,, .(3.9)
In the global coordinate, at the solution point kj
x,,
kj
j
kjx F
h
F,, )(
2
)(
ξ
=.(3.10)
The quantities kjx
F,
)( given by (3.9) and (3.10) are employed to approximate x
fat the solution point kj
x,for the
conservation law. Finally,
kjx
tt
kjkj F
dt
tdu
dt
du
n,
,, )(
)( −=






=
=
.(3.11)
The solution kj
u,can then be updated via an explicit Runge-Kutta method.
What is crucial in (3.9) is the following. At each solution point, the derivative kj
F,
)(
ξ
of the continuous flux
function is obtained by correcting the derivative kj
f,
)(
ξ
of the discontinuous flux function. The correction amount

American Institute of Aeronautics and Astronautics
9
is straightforward once the derivatives )(
LB k
g
ξ
′
and )(
RB k
g
ξ
′
are known for Kk ≤≤1. These derivatives, in turn,
can easily be derived once LB
gand RB
gare defined on
I
.
We summarize the algorithm for the new flux reconstruction schemes below.
Algorithm. At time level
n
, suppose the data kj
u,are known for all
j
and k.
(1) At each interface 2/1+j, if the left and right values Lj
u,2/1+and Rj
u,2/1+are not available, calculate
them; then obtain and store the upwind flux at all interfaces via (2.20) or (2.21).
(2) In the cell
j
, for Kk ...,,1=, evaluate the fluxes )( ,, kjkj uff =at the
K
solution points. Next, using
these fluxes, obtain the derivatives of the discontinuous flux function in the local coordinate, i.e., kj
f,
)(
ξ
,
by (2.11).
(3) At the two interfaces of cell
j
, calculate the jumps )1(
upw,2/1 −−
−jj ff and )1(
upw,2/1 jj ff −
+. For
Kk ...,,1=, evaluate the derivatives of the continuous flux function in the local coordinate namely,
kj
F,
)(
ξ
, by (3.9). Then obtain kjx
F,
)( by (3.10).
(4) Using the corrected slope kjx
F,
)( , calculate dtdu kj /
,by (3.11) and march in time by a Runge-Kutta
method. This completes the algorithm.
Next, we show that the resulting scheme is conservative, that is, we wish to show
upw,2/1upw,2/1
),( +−
=
−=






∂
∂jj
tt
Ejffdxtxu
tn
j
.(3.12)
To this end, consider the quadrature of degree of precision at least
1
−
K
using the
K
solution points, i.e., it is exact
for polynomial of degree
1
−
K
or less. Denote the weight at k
ξ
by k
w. Then, since )
ξ
(
j
uis of degree
1
−
K
,

=
1
1− =K
kkjkj uwdu 1,
)(
ξξ
.(3.13)
Note that the above is an equation, not an approximation. For the global coordinate,

=
=K
kkjk
j
Ejuw
h
dxxu
j1,
2
)( .(3.14)
The left hand side of (3.12) can then be written as

===
=
−==
∂
∂
=






∂
∂K
kkjxk
j
K
k
kj
k
j
K
kkjk
j
tt
EjFw
h
dt
du
w
h
uw
t
h
dxtxu
tn
j1,
1
,
1,)(
222
),( .(3.15)
Here, the last equation is a consequence of (3.11). Since xj
F)(is a polynomial of degree
1
−
K
, using the
quadrature formula (3.14),

=
=K
kkjxk
j
Exj Fw
h
dxxF
j1,
)(
2
)()( .(3.16)
By the fundamental theorem of Calculus,
upw,2/1upw,2/12/12/1 )()())(( −+−+ −=−=
jjjjjj
Exj ffxFxFdxxF
j
.(3.17)
The above three equations then imply (3.12). This completes the proof.
From here on, we focus only on LB
gsince RB
gfollows by symmetry. For simplicity of notation, when there is
no confusion, we drop the subscript and denote LB
gby
g
. Recall that a polynomial of degree
K
can have no more
than
K
zeros. The right interface 1
=
ξ
is a zero; as a result,
g
is completely defined if we specify
1
−
K
additional zeros. For examples, for the case
2
=
K
(piecewise linear or linear element),
g
is a quadratic defined by
(3.5) and one additional zero; for
4
=
K
(cubic element),
g
is of degree four and defined by again (3.5) and three
additional zeros. See Fig. 3.1(b).

American Institute of Aeronautics and Astronautics
10
IV. Choices of Correction Functions
This section employs the Legendre, Radau, and Lobatto polynomials and some of their main properties. Since
these polynomials and properties play a key role, they are reviewed in the Appendix. We also use several new
polynomials and properties, and these are derived in the Appendix as well.
Under the above approach of flux reconstruction, the problem of constructing high-order schemes reduces to
defining a correction function
g
on ]1,1[
−
=
I
, which is a polynomial of degree
K
determined by 1)1( =−g,
0)1( =g, and
1
−
K
additional conditions. These conditions are prescribed so that
g
approximates the function 0 in
some sense. But what criteria should we employ? The final criteria are the stability and accuracy of the resulting
scheme discussed in Section 6. Since there is generally a trade-off between accuracy and stability, and the property
of superaccuracy does not hold for the general case of the Euler or Navier-Stokes equations, an optimal scheme has
not been determined by this author. In fact, such a scheme is likely to be problem dependent. In this section,
therefore, we present various choices for
g
via approximation theory as well as some guidance from Fourier
analysis. To distinguish the several correction functions, they are assigned a subscript. We employ a subscript of
letters if the correction function leads to an existing scheme e.g., DG
gand SG
g(staggered-grid) or if it involves
well-known points, e.g., Lo
g(Lobatto), Ga
g(Gauss). We also use integer subscripts, e.g., K
gg ...,,
1. For a more
precise notation, we need two subscripts, e.g., K
gDG, or K
g,2 , but the subscript
K
for the number of solution points
is understood. Unless otherwise stated, we use the notations K
ξξ
...,,
1for the solution points, and 1
...,, −1 K
ζζ
, and
1=
K
ζ
for the zeros of the correction function.
The first choice for
g
, denoted by DG
gor 1
g(the right boundary is a zero of multiplicity one), is derived by the
following argument. Since
g
is of degree
K
and approximates the function 0, using the criterion of least squares,
we could require that the projection of
g
onto 1−K
Pis 0, where 1−K
Pis the space of polynomials of degree
1
−
K
or less. This requirement would imply that K
Pg ×= constant , where K
Pis the Legendre polynomial of degree
K
.
Since K
K
P)1()1( −=− and 1)1( =
K
P, the conditions 1)1( =−gand 0)1( =gcannot be met simultaneously no
matter which constant we choose. Therefore, the highest accuracy possible (via least squares) is the requirement that
the projection of
g
onto 2−K
Pis 0. This requirement is met by KR
R,, the right Radau polynomial of degree
K
shown in Fig. 4.1(a). The zeros of the right Radau polynomial are the right Radau points. Note that the correction
function for the left boundary is the right Radau polynomial, which vanishes at
1
=
ξ
.
The following remarks are in order. The requirement that
g
is perpendicular to 2−K
Pprovides
1
−
K
conditions
in determining
g
; the other two conditions are 1)1( =−gand 0)1( =g. As will be shown in Section 7, the resulting
scheme turns out to be identical to the DG method; for this reason, this correction function is denoted by DG
g.
Later, we will show via Fourier analysis that the scheme is accurate to order
1
2
−
K
, a fact consistent with the
super-accuracy of the DG scheme. The function DG
g(a more precise notation is K
g,DG ) can be expressed in terms
of the Legendre polynomial as
)(
2
)1( 1,,DGDG −
−
−
=== KK
K
KRK PPRgg .(4.1)
Since the Legendre polynomial K
Pis perpendicular to 1−K
P, the above expression for DG
gshows that it is
perpendicular to 2−K
P. While the slopes at the solution points must generally be evaluated numerically, the slopes
at the two boundaries are given bythe following formula:
2
)1()1()1(and
2
)1()1( 1
,DG
2
,DG K
Rg
K
Rg K
KRKR −
−=
′
=−
′
−=−
′
=−
′.(4.2)
The next correction function, denoted by SG
g,results in a simplified version of Kopriva and Kolias’ staggered-
grid scheme (1996). It is derived as follows. Consider the
1
+
K
Chebyshev-Lobatto points on
I
defined by, for
Kk ...,,0=,
)/cos( Kk
k
πζ
−= .(4.3)
The function SG
gis required to take value 1 at 1
0−=
ζ
and 0 at all other k
ζ
,Kk ...,,1=. See Fig. 4.1(b). The
resulting scheme is more economical than the original staggered-grid version since we only need one grid, namely,

American Institute of Aeronautics and Astronautics
11
the solution points; the Chebyshev-Lobatto (or flux) points are used only in the derivation of
g
and not in the
coding of the method. As will be shown in Section 6, compared to DG, the staggered-grid scheme is less accurate,
but to its advantage, the CFL limit is more than 3/2 times as large. Although the scheme had the disadvantage of
being mildly unstable, this problem will be remedied by one of the new methods that share the symmetry property
discussed below.
In general, if the
1
−
K
interior zeros of the correction function, namely k
ζ
, 1...,,1 −= Kk , are symmetric about
0, then both LB
gand RB
ghave the same interior zeros; therefore, )()( kjkj fF
ζζ
=where j
Fis given by (3.8).
As a result, the continuous flux function )(xFjcan be defined by the two interface upwind fluxes and the values
)( kj
f
ζ
, 1...,,1 −= Kk as in (Kopriva and Kolias 1996). The formulation here, again, is more economical since
these )( kj
f
ζ
are not needed.
As shown in Fig. 4.1(b), loosely put, SG
gis less steep than DG
g. Generally, a steeper correction function results
in a scheme with a smaller CFL limit.
Another difference between the current version and the original staggered-grid scheme is the following. For the
staggered-grid approach, one first interpolates the values kj
u,at the solution points to obtain the
u
values at the
interior flux points, and then, calculates the corresponding flux values at these flux points. For the current approach,
these flux values, though never explicitly evaluated, correspond to first calculating the fluxes at the solution points,
then interpolating these fluxes via the flux polynomial (discontinuous flux function) )(xf jto the flux points.
-
1
-
0.5
0
0.5
1
-
0.4
-
0.2
0
0.2
0.4
0.6
0.8
1
R
R
,1
R
R
,2
R
R
,3
R
R
,4
(a) -1 -0.5 0 0.5 1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
(b)
Figure 4.1. Correction functions.(a) The first choice for
g
denoted by DG
g or, more precisely, K
g,DG is the
right Radau polynomial KR
R,; here, we plot KR
R,for 4...,,1
=
K
. The resulting scheme yields a simplified version
of the DG scheme. (b) Correction function SG
g for the case
4
=
K
. The resulting scheme is a simplified version of
Kopriva and Kolias’ staggered-grid method. Note that the interior zeros of SG
g (the dots excluding the ones on the
boundaries) are symmetric about 0. Also shown is DG
g for comparison. Loosely put, SG
g is less steep than DG
g .
Generally, a steeper correction function results in a scheme with a smaller CFL limit.
We now discuss correction functions that result in new schemes.
The next
g
, denoted by 2
gor LoLump,
g(for ‘lumping for Lobatto points ’ explained later), is defined as follows.
Recall that DG
gis perpendicular to 2−K
P. If we relax this condition to the requirement that
g
is perpendicular to
3−K
P, then we have one degree of freedom to experiment with. Since a steeper correction function tends to result in
a scheme with a smaller CFL limit, we wish to make
g
less steep. On the other hand, since
g
approximates the
function 0 on
I
, it seems reasonable to require that all zeros of
g
lie on
I
. To make
g
less steep, therefore, the
extra condition is obtained by pushing one of the zeros to the right boundary. Thus,
g
is determined by the
conditions that it is perpendicular to 3−K
P, 1)1( =−g, 0)1( =g, and 0)1( =
′
g. Since
1
=
ξ
is a root (zero) of
multiplicity two, this correction function is denoted by 2
gand shown in Fig. 4.2(a).
The correction function 2
ghas the following remarkable property, which holds true for any
K
. Among the
K
Lobatto points (the zeros of the Lobatto polynomials of degree
K
),
′
2
gvanishes at
1
−
K
of them; the exception is
the left boundary point
−1
=
ξ
as shown in Fig. 4.2(a). Therefore, if we employ 2
g, it is convenient and
SG
g
ξ
DG
g
SG
g
DG
orsPolynomialRadau g
ξ
η
η

American Institute of Aeronautics and Astronautics
12
economical to select the
K
Lobatto points as solution points. The jump in flux values at the left interface results in a
correction to only 1,
)( jx
fand not to any kjx
f,
)( ,1>k. That is, the correction due to the jump at the left boundary
is lumped into that boundary, and the corrections at all other solution points are zero. For this reason we also use the
notation LoLump,
g. Also note that, 2/)1()1(
2−−=−
′
KKg , which equals 1
w−where 1
wis the weight at the
boundary point of the Lobatto quadrature with
K
points. This correction function can be expressed as a weighted
average of the Radau polynomials kR
R,:
1,,2
1
2
1
2
1−
−
+
−
−
=KRKR R
K
K
R
K
K
g.(4.4)
Via Fourier analysis for values
K
up to 10, it will be shown that 2
gresults in a scheme accurate to order
2
2
−
K
as opposed to order
1
2
−
K
of the DG scheme. To its advantage, the CFL limit is roughly twice as large as
that of DG.
-
1
-
0.5
0
0.5
1
-
0.25
0
0.25
0.5
0.75
1
1.25
(a) -1 - 0 . 5 0 0. 5 1
-0 . 5
0
0.5
1
(b)
Figure 4.2. Correction functions.(a) If we choose the K Lobatto points as solution points (small dots), then
′
2
g
vanishes at all solution points except the left boundary. In other words, the correction is lumped to the left
boundary. For this reason, 2
g is also denoted by LoLump,
g.As will be shown, the resulting scheme is accurate to
order
2
2
−
K
, one order lower than that of the DG scheme. (b) Lumping for the Chebyshev-Lobatto points with
,5,4
=
K
and 6. Note the large magnitude of the oscillations and the fact that
g
reaches local extrema at the
solution points.
We can apply the above lumping strategy to any set of solution points K
ξξ
...,,
1. Indeed, the correction function
g
, which is a polynomial of degree
K
, can be defined by requiring that 1)1( =−g, 0)1( =g, and 0)( =
′
k
g
ξ
for
Kk ,...,2=. That is, we lump the correction to the solution point closest to the left boundary, and )( 1
ξ
g
′
can easily
be calculated once
g
is determined. For a certain choice of solution points such as the Legendre-Lobatto points, this
requirement yields a stable and accurate scheme. However, in general, the resulting
g
may be a poor approximation
to the function 0, and the scheme may have large errors and/or be unstable.
In addition to the Legendre-Lobatto points, which lead to 2
g, we also test the Chebyshev-Lobatto points, i.e.,
)/cos( Kk
k
πξ
−= ,Kk ,...,1=. For this choice, the correction function that lumps the correction to the left
boundary has somewhat large oscillations: the minimum values for
g
are less than 7.0
−
. See Fig. 4.2(b).
We now derive a series of correction functions using an argument similar to that in the derivation of 2
g. Recall
that
g
is defined by the condition 1)1( =−gand
K
additional conditions. These
K
additional conditions are, for
DG1 gg =, the point
1
=
ξ
is a zero of multiplicity one, and 1
gis perpendicular to 2−K
P. For 2
g, the point
1
=
ξ
is
a zero of multiplicity two, and 2
gis perpendicular to 3−K
P. Following this pattern, for any integer 1
≥
m, in
addition to the condition 1)1( =−g, the correction function m
gis defined by the
K
conditions that
1
=
ξ
is a zero
of multiplicity
m
and m
gis perpendicular to 1−−mK
P. See Fig. 4.3(a). For the advection equation, it will be shown
via Fourier analysis that the corresponding scheme is (super) accurate to order mK
−
2.
pointsLobatto-ChebyshevwithLumping
ξ
5
=
K
ξ
2
g
η
4
=
K
6
=
K
pointsLobatto-LegendrewithLumping
η

American Institute of Aeronautics and Astronautics
13
If 1
−
=
Km , then 1−K
gis defined by the following conditions. First, the point
1
=
ξ
is a zero of
multiplicity
1
−
K
. Next, since 01
=
−
−
mK , we have 01 PP =
−− mK ; since 0
Pis the subspace (in this case, a line)
consisting of all constant functions, the condition that 1−K
gis perpendicular to 0
Pmeans
0()1,( 1
111 =)= −−−
ξξ
dgg KK . (4.5)
That is, 1−K
gis defined by 1)1(
1=−
−K
g,
1
=
ξ
is a zero of multiplicity
1
−
K
, and the above equation.
If Km
=
, then the
K
additional conditions for defining K
gare that the point
1
=
ξ
is a zero of multiplicity
K
. For this case,
K
K
g





2
−1
=)
ξ
ξ
(.(4.6)
Note that K
gis decreasing on
I
and 0≥)
ξ
(gfor all
ξ
in
I
.
-
1
-
0.5
0
0.5
1
-
0.4
-
0.2
0
0.2
0.4
0.6
0.8
1
(a)
-1
-0.5
0
0.5
1
-0.2
0
0.2
0.4
0.6
0.8
1
(b)
Figure 4.3. More correction functions.(a) Correction functions m
g,
K
m
...,,1
=
for the case
4
=
K
.For all
tested values of
K
( up to 10), m
g results in a scheme accurate to order mK
−
2.(b) Three correction functions
whose interior zeros are symmetric about
0
=
ξ
; they are SG
g (using the Chebyshev-Lobatto points), Lo
g
(Legendre-Lobatto), and Ga
g (Gauss). Here, Ga
g results in a stable scheme, which is accurate to order
2
2
−
K
.
The other two results in mildly unstable schemes, which are accurate to order
K
.
The following discussion on m
gis in order. First, the slope at the left boundary of m
gare given by, for
Km
≤
≤
1,
)1(
2
1
)1( +−−=−
′mKKgm.(4.7)
In particular, 2/)1( 2
1Kg −=−
′and 2/)1( KgK−=−
′
. The above can also be expressed recursively as follows.
With 2/)1( 2
1Kg −=−
′, for 11
−
≤
≤
Km ,
)2/()1()1(
1Kgg mm +−
′
=−
′
+.(4.8)
As an example, for the case
4
=
K
shown in Fig. 4.3(a), we have 8)1(
1−=−
′
g,6)1(
2−=−
′
g,4)1(
3−=−
′
g, and
2)1(
4−=−
′
g.
For all correction functions tested by this author, with
K
solution points, if
g
is of degree
K
and perpendicular
to m
Pwhere 20
−
≤
≤
Km , then the resulting scheme has order of accuracy 1
+
+
mK via Fourier analysis.
The next two correction functions are similar to SG
gof the staggered-grid scheme in that their interior zeros,
(i.e., excluding 1=
K
ζ
) are symmetric about
0
=
ξ
. For the first of the two, instead of the Chebyshev-Lobatto
points, consider the
1
+
K
Legendre-Lobatto points K
ζζ
...,,
0where 1−=
0
ζ
and 1=
K
ζ
.Let Lo
g(Legendre-
Lobatto) be defined by 1=− )1(gand
g
vanishes at K
ζζ
...,,
1. See Fig. 4.3(b). This correction function, similar to
SG
g, is not perpendicular to any Legendre polynomial k
P,Kk ...,,0=, i.e., Lo
gis not perpendicular to any k
P.
As will be shown, the resulting scheme is accurate to order
K
. It is also slightly more unstable than the staggered-
grid scheme.
ξ
η
Lo
g
Ga
gSG
g
1
g2
g3
g
4
g
ξ
η

American Institute of Aeronautics and Astronautics
14
The instability of the schemes corresponding to SG
gand Lo
gis remedied by the next correction function
denoted by Ga
g(Gauss) and defined as follows. Consider the
1
−
K
Gauss points (the zeros of the Legendre
polynomial 1−K
P). They are completely different from the
K
Gauss points (the zeros of the Legendre polynomial
K
P), which can be chosen as the
K
solution points. Recall that the quadrature associated with the
1
−
K
Gauss
points has a degree of precision of 321)1(2 −=−− KK . We define Ga
gby requiring that its interior zeros are these
1
−
K
Gauss points. It follows that Ga
gis perpendicular to 3−K
P. Indeed, recall that Ga
gis of degree
K
. Let
p
be a polynomial of degree 3
−
K, then Ga
gp is of degree 32
−
K. Since Ga
gp vanishes at the
1
−
K
Gauss points,
due to the degree of precision of the Gauss quadrature, 0((
1
1Ga =))
−
ξξξ
dgp , and the claim that Ga
gis
perpendicular to 3−K
Pfollows.
Since Ga
gis perpendicular to 3−K
P, the resulting scheme is (super) accurate to order
2
2
−
K
. To its advantage,
the CFL limit, as will be shown, is roughly 3/2 that of DG.
We conclude this section by the following observation. The zeros of Ga
glie closer to the boundaries than those
of SG
g, which, in turn, are closer to the boundaries than those of Lo
g(i.e., LO1,SGGa
ζζζ
<< 1,1, ). Consequently,
among these three correction functions, Ga
gis, loosely put, the steepest; it will be shown that the resulting scheme
has the smallest CFL limit.
V. Fourier (Von Neumann) Analysis
Fourier analysis provides information on both stability and accuracy of schemes. Concerning accuracy however,
it is worth keeping in mind that the super-accuracy property of schemes no longer holds for the general case of the
Euler or Navier-Stokes equations due to nonlinear errors.
For Fourier analysis, on the domain (−∞,∞), the conservation law is the advection equation
0=+ xt uu .(5.1)
The initial condition is periodic:
xwi
exu =)(
init (5.2)
where
w
is a real number between
π
−
and
π
called a wave number. Low frequency data corresponds to
w
of
small magnitude; high frequency, to
w
near
π
±
. The exact solution is
)(
exact ),( txwi
etxu −
=.(5.3)
At 0
=
xand 0
=
t, the exact time partial derivative is purely imaginary:
wiu t−=)0,0()( exact .(5.4)
The numerical schemes employ the cells ]2/1,2/1[ +−= jjE j. Denote the solution points in the local
coordinate on ]1,1[
−
=
I
by k
ξ
,Kk ...,,1=. The global solution points are
2/
,kkj jx
ξ
+= .(5.5)
The initial condition (5.2) evaluated at these solution points are ])2/([exp
,kkj jwiu
ξ
+= .(5.6)
However, it is not the above expression but the following property of the data, which plays a key role in the
calculation of eigenvalues,
kj
wi
kj ueu ,,1 −
−=. (5.7)
Note that to calculate the eigenvalues, the
K
solution points are grouped together as a vector, and we must
express the solution vector in terms of the data vectors. In this manner, the solution vector for the cell 1
−
j
looks
the same as that for the cell
j
except for a shift in the index.
Since the advection speed is 1, the wind blows from the left, and the upwind value at each interface 2/1+jis
the value from the cell
j
. As a result, there is no need to carry out the correction for the right boundary, i.e., we
only need LB
g, which is denoted by
g
. Next, in the local coordinates
ξ
on
I
, set

American Institute of Aeronautics and Astronautics
15










==










′
′
=
′










=
111
)(
)(
)(
)(
and
KKK g
g
g
g
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
MMM gg
(5.8)
That is, the vector of solution points is

, the vector of slope corrections at the solution points,
ξ
g. For the global
element, set










=










=
11
Kj
j
j
Kj
j
j
u
u
x
x
,
,
,
,
and MM ux (5.9)
Denote by land
r
the (column) vectors of
K
components whose dot product with j
uyield the interface values at
the left and right boundary, namely, 2/1−jand 2/1+j, respectively. In the local coordinate,
j
T
jkj
K
kkjj
T
jkj
K
kkj uruulu ururulul ======−  == .)1(and.)1( ,
1
,
1(5.10)
where the dot (.) represents dot product of two (column) vectors, the superscript Tdenotes the transpose, and the
space (e.g., j
Tul ) represents matrix multiplication. Let Dbe the derivative matrix defined in (2.12) and (2.14), and
let k
dbe the −kth row of D. Then
jkkj
uud .)()( =
ξ
ξ
.(5.11)
Since uf =and the wind blows from the left, )1(
1upw,2/1 −− =jj uf . Therefore, in the local coordinate, with j
F
denoting the continuous flux function,
ξξξξξ
guguuuF j
T
j
T
jjjjj )()()]1()1([)()( 11 ulur −+=−−+= −− . (5.12)
By evaluating the above at the solution point k
ξ
and, since
I
has length 2,
)]()(.[2 1
,kj
T
j
T
jk
kj g
dt
du
ξ
ξ
ulurud −+−= −.(5.13)
Or, in vector form, noting that )( 1j
T
j
Tulur −
−is a scalar quantity while Dis a matrix,
])([2 1
ξ
guluruD
uj
T
j
T
j
j
dt
d−+−= −.(5.14)
The above can be rearranged as follows. Since j
Tul is a scalar and
ξ
gis a vector,
j
T
j
T
j
Tulgulggul )()()(
ξξξ
== . (5.15)
Here,
ξ
gis a column vector; T
lis a row vector; as a result, T
lg
ξ
is a
K
K
×
matrix. Next, using (5.15), we
rearrange (5.14) as
}{ ])([)(2 1j
T
j
T
j
dt
dulgDurg
u
ξξ
−+−= −.(5.16)
Denote (
K
K
×
matrices)
])([2and2 01 TT lgDCrgC
ξξ
−−=−=
−. (5.17)
Then, (5.16) takes the form
jj
j
dt
duCuC
u011 += −− .(5.18)
Replacing 1−j
uby j
wi
eu
−as in (5.7), we obtain
j
wi
je
dt
duCC
u)( 01 += −
−.
Therefore, the discretization of space (semidiscretization) results in the equation (see also Chapter 10 of Hirsch
1988)
j
j
dt
duS
u=(5.19)
where the matrix S(for ‘space’ or ‘semidiscrete’) is defined by

American Institute of Aeronautics and Astronautics
16
01 CCS += −
−wi
e.(5.20)
Equation (5.19) is similar to the differential equation udtdu
λ
=/whose solution is t
ce
λ
where
c
is an arbitrary
constant. If 0)(Re ≤
λ
, the solution is stable; otherwise, it blows up as
t
tends to infinity. Here, the eigenvalues of S
take the place of
λ
and, for all schemes discussed here, Shas Keigenvalues. The eigenvalue which approximates
iw
−
(the exact eigenvalue of the operator
x
∂
−∂
/given by (5.4)) is called the principal eigenvalue and is denoted
by )(wS :
iwwS −≈)(. (5.21)
All other eigenvalues are spurious ones. All eigenvalues must lie in the left half of the complex plane for the
semidiscretization to be stable. If any of the eigenvalues lies in the right half of the complex plane ( 0)(Re >
λ
), the
scheme may eventually blow up unless the instability is mild and the time stepping provides some additional
dissipation to stabilize the scheme. As will be shown below, some of the existing semi-discrete schemes are mildly
unstable, but in practical calculations, the fully discrete versions are stable: the schemes convect the waves out of the
domain of calculation long before any instability occurs.
To find the order of accuracy of a scheme, one can expand theprincipal eigenvalue )(wS as a Taylor series in w.
A scheme is m-th order accurate if, for small w,
)()( 1+
+−= m
wOiwwS .(5.22)
Note that Sis an approximation to the scaled derivative hux
−where his the cell width.
In practice, it is difficult if not impossible to derive an expression for )(wS when the number of solution points
is greater than 2. Therefore, we obtain the order of accuracy of a scheme bythe following approximation. First, let w
be fixed, say, 4/
π
=w. We can estimate the error
iwwSwE += )()(. (5.23)
Next, by halving the wave number w(which is equivalent to doubling the number of mesh points), the error
corresponding to w/2 (say, 8/2/
π
=w) is 2/)2/()2/( iwwSwE += .(5.24)
Since
)(
2
1
21
11 +
++






≈












m
mm
wO
w
O,
for a scheme to be m-th order accurate,
)(
2
1
)2/( 1wEwE m+






≈.
That is,
1
2
)2/(
)( +
≈m
wE
wE .
Taking the logarithm of both sides of the above, the order of accuracy is given by the integer
1)2(Log
)2/(
)(
Log −











≈wE
wE
m.(5.25)
We now show that the eigenvalues of Sdefined by (5.19) and (5.20) are independent of the solution points
chosen. To proceed, consider the local coordinate
ξ
on ]1,1[
−
=
I
. Let k
ξ
be the solution points and k
uthe
corresponding data. Then =)
K
kkk
u
1(
ξφ
is the polynomial of degree
1
−
K
interpolating the data. Denote the
vector K
kk
u1
}{ =by
u
. Let l
ξ
~
,Kl ...,,1=be another set of solution points, and let l
u
~
be the interpolated value at l
ξ
~
,
K
ll
K
klkkl uuu 1
1}
~
{
~
and
~
(
~=
==)= u
ξφ
.(5.26)
For Klk ≤≤ ,1 , set
}{and
~
(kllkkl mm =)= M
ξφ
. (5.27)
Note that
M
is a
K
K
×
matrix. By (5.26) and (5.27),

American Institute of Aeronautics and Astronautics
17
Muu =
~
.(5.28)
Since we can obtain
u
from u
~
,
M
is invertible. Next, equation (5.19) in the local coordinate takes the form, using
the same notation as (5.19) for the sake of simplicity, uSu =dtd /. (5.29)
For the solution points l
ξ
~
, the corresponding equation is
uSu
~
~
/
~
=dtd .(5.30)
We wish to show that S
~
and Sare similar, and thus, share the same eigenvalues. To this end, multiplying (5.29)
on the left by
M
, we have
uMSuM =dtd /. (5.31)
Therefore, since
M
is invertible and independent of
t
,
))((/)( 11 uMMSMuMMSMuM −− ==dtd .(5.32)
That is, by (5.28),
uMSMu
~
)(/
~
1−
=dtd .(5.33)
Comparing (5.30) and the above, we obtain 1
~
−
=MSMS .(5.34)
That is, S
~
is similar to S, and the proof is completed.
Note that the derivative matrix follows the same transformation rule, namely, 1
~
−
=MDMD . As for T
ldefined
by (5.10), the transformation rule is 1
~
−
=Mll TT , whereas for
ξ
gin (5.8),
ξξ
gMg =
~
. We omit the details.
We conclude this section with a discussion on eigenvectors and superaccuracy. Assume that the initial condition
is xwi
exu =)(
init , or if we wish to have a real function, )Re()(cos)(
init xwi
ewxxu == . To start the calculation, we
define }...,,1,{ ,Kku kjj ==uby evaluating at the solution points,
K
kkj jwi 1
}])2/([{exp =
+=
ξ
u. (5.35)
For each cell j
E, the mapping from )(
init xu to 
=)()( ,, xuxu kjkjj
φ
can be considered as a ‘projection’ of the
initial condition onto 1−K
P. The error )()(
init xuxu j
−is of order )( K
wO (i.e., )( K
hO ). Thus, if we calculate the
error by comparing the solution )(xu jat the final time with exact
udefined in (5.3), the scheme is accurate to order
no higher than
K
.
One way to observe the property of super-convergence or super-accuracy numerically is as follows. Let Sbe the
matrix representing the spatial discretization defined by (5.20). For the schemes discussed here, the eigenvectors of
Sdenoted by k
eare independent: K
kk 1
}{ =
eform a basis for K
R
. Let 1
ebe the principal eigenvector corresponding
to the principal eigenvalue )(wS (a more precise notation is )(
1we). Since 0
uu jwi
je=, we only need to focus on
0
=
j
. The vector 0
ucan be decomposed using the basis K
kk 1
}{ =
eby
=
=K
kkk
c
1
0eu .(5.36)
The principal component is 11ec, which corresponds to the principal eigenvalue )(wS . This component is
propagated with the accuracy of the scheme, i.e., of order
m
as discussed above if the scheme is (super) accurate to
order
m
. For this component, the error accumulates with the number of time steps. Thus, one way to observe the
super-convergence or super-accuracy property is to use the principal eigenvalue 11ece wji as start-up solution and
calculate the error by comparing the final solution with this start-up solution.
VI. Stability and Accuracy of Schemes via Flux Reconstruction
As discussed in the previous section, Fourier stability and accuracy analyses of schemes are independent of the
solution points chosen. In other words, whether the Gauss or Radau or Lobatto or equidistant points are selected for
the solution points, the results from Fourier analysis are the same. These results vary only with the choice of

American Institute of Aeronautics and Astronautics
18
correction function. For brevity, we often identify the scheme by its correction function, e.g., scheme Ga
grefers to
the semidiscrete flux reconstruction scheme corresponding to the correction function Ga
g.
The case
1
=
K
. The corresponding correction function is )/2−=
ξ
1(g. Equation (5.18) then takes the form of
the first order upwind scheme:
jj
juu
dt
du −= −1.
The eigenvalue is
)(2/1)( 32 wOwiwewS wi +−−=−= −.
As
w
varies on ]2,0[
π
, )(wS varies on a circle of radius 1 centered at i01
+
−
on the complex plane, which is
called the spectrum of the semidiscrete scheme.
Note that for this correction function, 1( =)
1
1−
ξξ
dg ; thus,  )≠)
jj Ej
EjdxxFdxxf (( .
The case
2
=
K
. The correction functions are shown in Fig. 6.1(a). They are
4
1
2
4
and,
2
2
,
4
1
2
4
3LoLump,2GaLoSGDG +−==−===−−= 222
ξξξξξξ
gggggg .
-
1
-
0.5
0
0.5
1
-
0.2
0
0.2
0.4
0.6
0.8
1
DG
gSG
g
2
g
(a)
-
6
-
5
-
4
-
3
-
2
-
1
Re
1
2
3
Im
DG SG
(b)
-2
-1.5
-1
-0.5
Re
0.25
0.5
0.75
1
1.25
1.5
1.75
Im
(c)
Figure 6.1. The case
2
=
K
.(a) Correction functions.(b) Spectra of semi-discrete schemes (collection of all
eigenvalues). Due to symmetry, only the top half of the complex plane is shown. The spectrum of scheme 2
g turns
out to be identical to the boundary of the stability region of the explicit 2-stage Runge-Kutta time stepping method.
The maximum (artificial) dissipation occurs at 2
2−=xfor scheme 2
g (lumping for Lobatto points), 4
SG −=xfor
staggered-grid, and 6
DG −=x for DG. (c) Plots of (CFL limit)*spectrum. If we use explicit 2-stage Runge-Kutta
time stepping, the CFL limit for DG scheme is 3/1/ DG2 =xx ; that for SG, 2/1/ SG2 =xx ; and for scheme 2
g , 1.
Note that the curves (spectrum* CFL limit) lie inside the stability region of the 2-stage R-K scheme, which is the
region bounded by curve 2
g.
(
2
=
K
) Scheme Order of accuracy Coarse mesh error, 8/
π
=wFine mesh error, 16/
π
=w
DG 3i
54 103.3102.3 −− ×−×− i
65 101.1101.2 −− ×−×−
SG 2 i
34 104.2101.7 −− ×+×− i
45 101.3106.4 −− ×+×−
2
g=LoLump,
g2i
33 10.9105.2 −− ×+×− i
34 104.2101.7 −− ×+×−
Table 6.1. Orders of accuracy and errors of schemes for
2
=
K
. All errors have negative real parts. A closer
inspection shows that all eigenvalues lie on the left half of the complex plane; as a result, these schemes are stable.
The spectra (collection of all eigenvalues) of the corresponding semidiscrete schemes are shown in Fig. 6.1(b).
Due to symmetry, only the top half of the complex plane is shown. Each spectrum intersect the real axis at a location
corresponding to the maximum numerical dissipation of the scheme: at 2
2−=xfor 2
g,4
SG −=xfor the
staggered-grid, and 6
DG −=xfor DG scheme. These locations determine the CFL limits (the maximum xta ∆∆ /||
before the scheme blows up, where
a
is the wave speed, t
∆
time step size, and x
∆
mesh width).
Note the following surprising coincidence. For
2
=
K
, the spectrum of the scheme 2
g(spatial discretization)
turns out to be identical to the boundary of the stability region of the explicit 2-stage Runge-Kutta time stepping
method. This coincidence holds true for all
K
tested by this author: the spectrum of the scheme K
gturns out to be
identical to the boundary of the stability region of the explicit time stepping method with amplification factor
2
g
{Spectrum}*limit)(CFL
η
ξ
2
g
{DG}*(1/3)
SG}{*)2/1(

American Institute of Aeronautics and Astronautics
19
K
zz +++ ...1 where
z
is a complex number. (The
K
-stage Runge-Kutta method has such an amplification factor
for
4
≤
K
.)
If the 2-stage Runge-Kutta time stepping is employed, then the CFL limit equals 1 for scheme 2
g;2/1 for SG;
and 1/3 for DG. In addition, the curves (CFL limit) *spectrum shown in Fig 6.1(c) lie inside the region bounded by
curve 2
g.
The orders of accuracy and errors of the semi-discrete schemes for
2
=
K
are tabulated in Table 6.1. Observe
that the errors for all schemes have negative real parts, i.e., the eigenvalues lie on the left half of the complex plane.
A closer inspection shows that all eigenvalues share this property; therefore, these schemes are stable.
The case 3
=
K. The correction functions are shown in Fig. 6.2 where DG1 gg =,LoLump,2 gg =, and 3
gare
shown in (a) and SG
g,Lo
g, and Ga
g, in (b). Note that for each of the latter three, the interior zeros are symmetric
about 0. Also note that for this
K
,2LoLump,Lo-ChLump, ggg == .
For 3
=
K, the spectra (collection of all eigenvalues) of the corresponding semidiscrete schemes are shown in
Fig. 6.3(a). Similar to the case
2
=
K
, note that the spectrum of scheme 3
gturns out to be identical to the boundary
of the stability region of the explicit 3-stage Runge-Kutta method.
-
1
-
0.5
0
0.5
1
-
0.4
-
0.2
0
0.2
0.4
0.6
0.8
1
(a)
-
1
-
0.5
0
0.5
1
-0.2
0
0.2
0.4
0.6
0.8
1
(b)
Figure 6.2. Correction functions for the case 3
=
K;(a) DG1 gg =,LoLump,2 gg =(lumping for Lobatto points),
and 3
g;(b) SG
g,Lo
g,and Ga
g;the interior zeros for each of these three correction functions (dots) are
symmetric about 0.
-
12
-
10
-
8
-
6
-
4
-
2
Re
1
2
3
4
5
6
7
Im
(a)
-2.5
-2
-1.5
-1
-0.5
Re
0.5
1
1.5
2
Im
(b)
Figure 6.3. (a) Spectra of semidiscrete schemes for the case 3
=
K.(b) Plots of (CFL limit)*spectrum. If we
use the explicit 3-stage Runga-Kutta time stepping for these schemes, the CFL limit for the DG scheme is about .21
(slightly smaller than DG3 /xx ); for SG, .34; and for 2
g , .45. Note that the curves (CFL limit) *spectrum lie inside
the region bounded by curve 3
g , which is identical to the stability region of the 3-stage Runge-Kutta scheme.
Next, we discuss CFL limits. First, as shown in Fig. 6.3, the spectra intersect the real axis at: 8.11
DG −≈xfor
DG, 1.7
SG −≈xfor staggered-grid, 4.5
2−≈xfor scheme 2
g(lumping for Lobatto points), and 5.2
3−≈xfor
scheme 3
g. If the 3-stage Runge-Kutta time stepping is employed, then for DG, the CFL limit is (slightly smaller
ξ
ξ
1DG gg =2
g
3
g
Lo
g
Ga
g
SG
g
Ga
g
SG
Lo
g
LoLump,
g
3
g
}{*45. LoLump,
g
}SG{*34.
{DG}*.21
2
or g
DG
3
g
η
η
}{*32. Ga
g

American Institute of Aeronautics and Astronautics
20
than DG3 /xx ) about .21, SG .34, Ga
g.32, and LoLump,
g.45. Fig. 6.3(b) shows the plots of (CFL limit)*spectrum
for the schemes DG, SG, LoLump,
g, and Ga
g. Note that these curves lie inside the region bounded by the curve 3
g,
which is identical to the stability region of the 3-stage Runge-Kutta time stepping scheme.
(3
=
K)
Scheme Order of
accuracy Coarse mesh error, 8/
π
=wFine mesh error, 16/
π
=w
DG 5i
87 104.310.5 −− ×−×− i
109 107.2109.7 −− ×−×−
Ga
g4i
66 105.8104.1 −− ×+×− i
78 107.2102.2 −− ×+×−
LoLump,2 gg =4i
56 109.1102.3 −− ×+×− i
78 10.610.5 −− ×+×−
SG 3 i
55 104.1109.3 −− ×+× i
76 104.4105.2 −− ×+×
Lo
g3i
55 107.1104.6 −− ×+× i
76 104.5101.4 −− ×+×
3
g3i
44 101.3103.9 −− ×+× i
65 108.9101.6 −− ×+×
Table 6.2. Orders of accuracy and errors of schemes with 3 solution points ( 3
=
K). The errors for schemes
DG, Ga
g , and LoLu,
ghave negative real parts. A closer inspection of all eigenvalues shows that they lie on the left
half of the complex plane, and these three schemes are stable. On the other hand, errors for the three schemes SG
g ,
Lo
g , and 3
g have positive real parts; thus, these schemes are unstable; for SG
g and Lo
g , however, the instability
is mild.
The orders of accuracy and errors of the semi-discrete schemes for 3
=
Kare tabulated in Table 6.2. The errors
for schemes DG, Ga
g, and LoLu,
ghave negative real parts; a closer check of the eigenvalues reveals that these
three schemes are stable. On the other hand, errors for the three schemes SG
g,Lo
g, and 3
ghave positive real parts;
thus, these schemes are unstable; for the schemes SG
gand Lo
g, however, the instability is mild.
The next remark holds for all schemes and arbitrary
K
. Between the real and imaginary parts of the errors, if the
order of accuracy is odd, the leading error is real; otherwise, it is imaginary.
Finally, for the staggered-grid scheme, the maximum value of the real parts of the eigenvalues is about 0.0028,
which shows that the instability is mild. For this scheme, the unstable component is usually convected out of the
domain of calculation before it has a chance to contaminate the solution.
-1 -0.5 0 0.5 1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
(a) -1 -0.5 0 0.5 1
-0.2
0
0.2
0.4
0.6
0.8
1
(b)
Figure 6.4. Correction functions for the case
4
=
K
;(a) DG1 gg =,LoLump,2 gg =,3
g,4
g,and
Lo-ChLump,
g(lumping for Chebyshev-Lobatto points); (b) SG
g,Lo
g,and Ga
g;the interior zeros for each of these
three correction functions (dots) are symmetric about 0.
The case
4
=
K
. The correction functions are shown in Fig. 6.4 where DG1 gg =,LoLump,2 gg =,3
g,4
gand
Lobatto-ChebyshevLump,Lo-ChLump, gg =are shown in (a) and SG
g,Lo
g, and Ga
g, in (b). For each of the last three, the
interior zeros are symmetric about 0.
For
4
=
K
, the spectra (collection of all eigenvalues) of the corresponding semidiscrete schemes are shown in
Fig. 6.5(a). Similar to the cases of
2
=
K
and 3
=
K, the spectrum of scheme 4
gis identical to the boundary of the
stability region of the explicit 4-stage Runge-Kutta method.
ξ
DG
g
2
g
3
g
Lo
g
Ga
g
SG
g
4
g
1
g
Lobatto-ChebyshevLump,
g
ξ
η
η

American Institute of Aeronautics and Astronautics
21
(a)
-2.5
-2
-1.5
-1
-0.5
Re
0.5
1
1.5
2
2.5
3
Im
(b)
Figure 6.5. (a) Spectra of semidiscrete schemes with
4
=
K
.(b) Plots of (CFL limit)*spectrum. If we use the
explicit 4-stage Runge-Kutta time stepping for these schemes, the CFL limit for the scheme DG is about .21; for SG,
.34; and for 2
g , .45. Note that the curves (CFL limit)*spectrum lie inside the stability region of the 4-stage R-K
scheme, which is the region bounded by curve 4
g.
Next, we discuss CFL limits. First, the spectra intersect the real axis at: 2.19
DG −≈xfor DG,
1.14
Lo-ChLump, −≈xfor scheme Lo-ChLump,
g(lumping for Chebyshev-Lobatto points), 3.12
Ga −≈xfor Ga
g,
8.10
SG −≈xfor staggered-grid, 6.9
2−≈xfor 2
g(lumping for Lobatto points), and 8.2
4−≈xfor 4
g. If the 4-
stage Runge-Kutta time stepping is employed, then in increasing order, the CFL limits are, for DG, 145./ DG4 ≈xx ;
for Lo-ChLump,
g, .198; for Ga
g, .227; for SG, 257./ SG4 ≈xx ; and for LoLump,
g, .289. Thus, the CFL limit for
LoLump,
gscheme is about twice that for DG for the case
4
=
K
.
Fig. 6.5(b) shows the plots of (CFL limit)*spectrum for the following schemes: DG, SG, LoLump,
g,Lo-ChLump,
g,
and Ga
g. Note that these curves lie inside the stability region of the 4-stageRunge-Kutta time stepping scheme.
(
4
=
K
)
Scheme Order of
accuracy Coarse mesh error, 4/
π
=wFine mesh error, 8/
π
=w
DG 7i
87 10.110.1 −− ×−×− i
1110 10.210.4 −− ×−×−
Ga
g6i
67 103.1101.3 −− ×+×− i
89 101.1102.1 −− ×+×−
LoLump,2 gg =6i
67 103.2104.5 −− ×+×− i
89 109.1102.2 −− ×+×−
Lo-ChLump,
g4i
55 108.9103.3 −− ×+×− i
67 102.3103.5 −− ×+×−
SG 4 i
55 102.2102.1 −− ×−× i
77 108.710.2 −− ×−×
3
g5i
55 105.2109.5 −− ×+× i
77 10.2109.9 −− ×+×
Lo
g4i
55 109.3101.2 −− ×−× i
67 103.1104.3 −− ×−×
4
g4i
33 10.2105.1 −− ×−× i
55 104.7105.2 −− ×−×
Table 6.3. Orders of accuracy and errors of schemes for
4
=
K
. The errors for schemes DG, Ga
g , 2
g (lumping
for Lobatto points), and Lo-ChLump,
g(lumping for Chebyshev-Lobatto points) have negative real parts; these four
schemes are stable. On the other hand, errors for the four schemes SG, 3
gLo
g , and 4
g have positive real parts;
these schemes are unstable, but the instability of schemes SG and 3
g are mild.
The orders of accuracy and errors of the semi-discrete schemes for
4
=
K
are tabulated in Table 6.3. The errors
for schemes DG, Ga
g,2
g(lumping for Lobatto points), and Lo-ChLump,
g(lumping for Chebyshev-Lobatto points),
have negative real parts. A closer check of all eigenvalues reveals that these four schemes are stable. The errors for
the last four schemes, namely, SG, 3
g,Lo
g, and 4
ghave positive real parts; as a result, they are unstable. Among
-15
-10
-5
Re
2
4
6
8
10
12
Im
Ga
g
SG
Lo
g
LoLump,
g3
g
}{*289. LoLump,
g
}{*198. Lo-ChLump,
g
{DG}*.145
2
or g
DG
4
g
Lo-ChLump,
g4
g
SG}{*257.
}{227. Ga
g

American Institute of Aeronautics and Astronautics
22
-60
-50
-40
-30
-20
-10
Re
5
10
15
20
25
30
35
Im
Figure 6.7. Spectra of schemes with 8
=
K.
these four, the staggered-grid scheme has the mildest instability and the smallest error, whereas the scheme 4
ghas
the most severe instability and the largest error. For this reason, if 4
gwere to be employed, it must be blended with
other stable schemes, possibly in a manner similar to the useof unstable stencils in the ENO and WENO methods.
Finally, for each scheme, the real parts of the eigenvalues reach a maximum: for the staggered-grid scheme, this
maximum is about 0.015; for 3
g, 0.011; for Lo
g, 0.050, and for 4
g, 0.237.
The case 8
=
K. The correction functions are shown in Fig. 6.6; DG1 gg =,LoLump,2 gg =,3
g, and
Lobatto-ChebyshevLump,Lo-ChLump, gg =in (a); SG
gand Ga
gin (b). For the last two correction functions, the interior
zeros are symmetric about 0.
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
(a)
-1
-0.5
0
0.5
1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
(b)
Figure 6.6. Correction functions for 8
=
K;(a) DG1 gg =,LoLump,2 gg =,3
g, and Lo-ChLump,
g;(b) SG
g and
Ga
g.
The spectra of the semidiscrete schemes for 8
=
K
are shown in Fig. 6.7.
The orders of accuracy and errors of the semi-
discrete schemes for 8
=
Kare tabulated in Table 6.4.
Due to round-off errors, numbers less than 12
10−are
meaningless. The errors for schemes DG, Ga
g,2
g
(lumping for Lobatto points), and Lo-ChLump,
g
(lumping for Chebyshev-Lobatto points) have negative
real parts; these four schemes are stable. The errors for
schemes SG and 3
ghave positive real parts; these
schemes are mildly unstable. The maximum value of
the real parts of the eigenvalues is 0.000040 for SG and
0.00011 for 3
g.
Remaks for arbitrary
K
.
Concerning the instability of the staggered grid schemes, the maximum value of the real parts of the eigenvalues
are listed for
12
≤
K
in Table 6.5. Note that the larger the value, the more unstable the scheme. The scheme
corresponding to 8
=
Kis least unstable: the maximum value is only 0.000041.
The schemes DG, Ga
g, and LoLump,
gare stable for all values
K
tested by this author. Thus, it appears that if
g
is perpendicular to 3−K
P, then the resulting scheme is stable. The inverse, however, does not hold: the
scheme Lo-ChLump,
gis not perpendicular to any m
P, and it is stable. The following problems remain open: (1) What
are the necessary and sufficient conditions on the correction function for the resulting scheme to be stable? (2) Can a
scheme be optimized in a manner that it is mildly unstable (say the maximum real parts of the eigenvalues are less
than 0.001) but the CFL limit is largest possible?
Concerning the relation between the correction function and super-accuracy, from the above Fourier analysis, if
g
is perpendicular to m
P, then the resulting scheme is super-accurate to order 1
+
+
mK .
ξ
DG1 gg
=
2
g
3
g
Ga
g
SG
g
Lobatto-ChebyshevLump,
g
ξ
DG
Lo-ChLump,
g
Ga
g
SG
2
g
3
g
η
η

American Institute of Aeronautics and Astronautics
23
(8
=
K)
Scheme Order of
accuracy Coarse mesh error,
π
9.
=
wFine mesh error,
π
45.
=
w
DG 15? i
1110 109.1101.1 −− ×−×− 12
10|error| −
<
Ga
g?14 i
1010 104.9108.3 −− ×+×− 12
10|error| −
<
LoLump,2 gg =14? i
910 102.1109.4 −− ×+×− 12
10|error| −
<
Lo-ChLump,
g8i
77 102.2103.4 −− ×+×− i
910 103.1108.6 −− ×+×−
SG 8 i
86 104.410.1 −− ×+× i
99 105.1105.1 −− ×−×
3
g13? i
98 103.9103.1 −− ×+× 12
10|error| −
<
Table 6.4. Orders of accuracy and errors of schemes for 8
=
K.Note that due to round-off errors, numbers less
than 12
10−are meaningless. The errors for schemes DG, Ga
g , 2
g (lumping for Lobatto points), and Lo-ChLump,
g
(lumping for Chebyshev-Lobatto points) have negative real parts; these four schemes are stable. The errors for
schemes SG and 3
g have positive real parts; these schemes are mildly unstable. The maximum positive real parts
are: 0.000040 for SG and 0.00011 for 3
g.
K
3 4 5 6 7 8 9 10 11 12
Max. Value 0.028 0.015 0.028 0.031 0.011 0.000041 0.00037 0.0016 0.0041 0.0082
Table 6.5. The maximum value of the real parts of the eigenvalues for the staggered-grid schemes with
various
K
.
VII. Flux Reconstruction and DG Approaches
DG scheme via Flux Reconstruction Approach.
Next, we prove that the DG method can be cast as a flux reconstruction scheme: it is identical to the scheme
using the Radau polynomials as correction functions. To this end, we first rederive the DG scheme for the
conservation law 0=+ xt fu .
On the cell (or element) j
EE =, let the inner product of any two functions
v
and
w
be defined by

=Edxxwxvwv )()(),( .(7.1)
Formally, we require the solution
u
to satisfy, for any test function (polynomial of degree
1
−
K
)
φ
,
0=+
∂
∂
 Ex
Edxfdxu
t
φφ
.(7.2)
Applying integration by parts to the second term above, we obtain
[]
0=−+
∂
∂
 ∂Ex
E
Edxffdxu
t
φφφ
.(7.3)
For the DG method, the function
u
is replaced by the polynomial )(xu jof degree
1
−
K
,
φ
by one of the basis
functions kj,
φ
,Kk ...,,1=, and fby the piecewise polynomial )(xf jalso of degree
1
−
K
. As for the boundary
term, if ],[ 2/12/1 +−
== jjj xxEE , then
[
]
))(())(( 2/12/1 −+
∂−= jj
Exfxff
φφφ
. Since the piecewise polynomial
function }{ j
uis discontinuous across cell interfaces, so is the piecewise polynomial }{ j
f. Which value of f
should we employ to evaluate the boundary term? It is here where the upwind fluxes upw,2/1−j
fand upw,2/1+j
fplay
their role. The value of
φ
at the boundary, however, is that from the cell
E
(i.e., no upwinding). Again, the upwind
flux provides coupling (interaction) between the unknowns in adjacent cells and results in a conservative scheme.
Denoting the upwind flux by upw
f, the solution
u
is required to satisfy
[]
0
upw =−+
∂
∂
 ∂Ex
E
Edxffdxu
t
φφφ
.(7.4)

American Institute of Aeronautics and Astronautics
24
For the DG scheme, the solution is obtained by evaluating the above numerically. Here, however, we wish to
eliminate the test function
φ
. Applying integration by parts once more to Exdxf
φ
, since (the polynomials) f
and
φ
are smooth on
E
,
[]
 −= ∂Ex
E
Exdxffdxf
φφφ
.(7.5)
For the boundary term, in this case, both
φ
and fare calculated from the data inside the cell
E
(i.e., no
upwinding). Substitute this expression into (7.4), we obtain,
0])([ upw =+−+
∂
∂ ∂ExE
Edxfffdxu
t
φφφ
. (7.6)
Note that )( upw ff −in the boundary term is simply the jump at the interface in (3.4). The above, loosely put, is the
result of integrating by parts twice. It is simply the inner product of the conservation law 0=+ xt fu with the test
function
φ
on
E
, except for the correction term
[
]
E
ff ∂
−
φ
)( upw . This term takes into account the upwind effect
(the interaction of the data between adjacent cells).
Next, we switch to the local coordinate
ξ
on ]1,1[
−
=
I
. Denote the length of
E
by h. Noting that
ξ
dhdx )2/(=and
ξ
ξ
dfdxfx=, in the local coordinate, Eq. (7.6) takes the form
0)1(])1()1([)1(])1()1([
2
1
1
upwupw
1
1=+−−−−−−+
∂
∂ −−
ξφφφξφ
ξ
dfffffdu
t
h.(7.7)
The next technique is crucial since it helps eliminate the test function
φ
. Focusing first on the left boundary
1
−
=
ξ
, in view of the expression −
1
1
ξφ
ξ
df above, the following question can be raised: can we find a polynomial
LB
gon ]1,1[
−
=
I
which possesses the property that for any polynomial
φ
of degree
1
−
K
or less,
−′
=−− 1
1LB
)1(
ξφφ
dg .(7.8)
Here, )=)
′
ξξ
ξ
()(( LBLB gg . From (7.7), we require )
′
ξ
(
LB
gto have the same degree as )
ξ
(
j
u, i.e., of degree
1
−
K
; as a result, LB
gis required to be of degree
K
. Applying integration by parts to the above right hand side, we
have
−′
−−−−=−− 1
1LBLBLB )1()1()1()1()1(
ξφφφφ
dggg .(7.9)
Thus, Eq. (7.8) holds if LB
gsatisfies,
0)1(,1)1( LBLB ==− gg ,(7.10)
and, for all
φ
in 1−K
P,
0
1
1LB =
′
−
ξφ
dg .(7.11)
Since
φ
is of degree
1
−
K
,
φ
′
is of degree
2
−
K
; moreover,
φ
′
spans 2−K
Pas
φ
spans 1−K
P. The above then
implies that LB
gis orthogonal to 2−K
P, i.e., for any polynomial
ϕ
of degree
2
−
K
,
0
1
1LB =
−
ξϕ
dg .(7.12)
The criterion that LB
gis orthogonal to 2−K
Pprovides
1
−
K
conditions in determining LB
g. The other two are
provided by (7.10). These
1
+
K
conditions imply that LB
gis the right Radau polynomial on
I
(see the Appendix):
),()(
LB
ξξ
KRg R
=.(7.13)
Again, note that the correction function for the left boundary is the right Radau polynomial, which vanishes at the
right boundary 1
=
ξ
.
Another way to look at LB
gdefined by (7.8) is by using distribution theory. Equation (7.8) implies that
′
−LB
g
is an approximation to the Dirac delta function at 1
−
=
ξ
:

American Institute of Aeronautics and Astronautics
25
)1(
LB −≈
′
−
δ
g.
Since the integral of the Dirac delta function is the unitstep (or Heaviside) function,



−
≤
=+≈+− otherwise.1
,1if0
)1(constant
LB
ξ
ξ
Hg
By choosing the constant to be 1, after a little algebra, the above implies that LB
gapproximates the ‘step-down
function’ 1−
hdefined on
I
by



−>
−=
=
−.1if0
,1if1
)(
1
ξ
ξ
ξ
h(7.14)
Now, we switch to the right boundary. In a manner similar to (7.10) and (7.12), let RB
gbe a polynomial of
degree
K
defined by 1)1(,0)1( RBRB ==− gg ,(7.15)
and, for all
ϕ
in 2−K
P,
0
1
1RB =
−
ξϕ
dg .(7.16)
Then RB
gis left Radau polynomial, which vanishes at the left boundary
−1
=
ξ
,
),()(
RB
ξξ
KRg L
=.(7.17)
Let the ‘step-up’ function 1
hbe defined on
I
by



=
<
=.1if1
,1if0
)(
1
ξ
ξ
ξ
h(7.18)
Then RB
gapproximates 1
h. For any
φ
in 1−K
P,
φ
′
is in 2−K
P. Replacing
ϕ
by
φ
′
for the left hand side of (7.16)
and applying integration by parts, we obtain
 −− ′
−−−−=
′1
1RBRBRB
1
1RB )1()1()1()1(
ξφφφξφ
dgggdg .(7.19)
Using (7.15) and (7.16), the above implies
−′
=1
1RB
)1(
ξφφ
dg .(7.20)
We now return to our main task of eliminating
φ
. By (7.20) and (7.8), Eq. (7.7) can be written as
[][ ]
0)1()1()1()1(
2LBupwRBupw =+
′
−−−+
′
−+
∂
∂ IIII dfdgffdgffdu
t
h
ξφξφξφξφ
ξ
. (7.21)
What is crucial here is that we can factor
φ
out:
{}
0)]1()1([)]1()1([
2LBupwRBupw =−−−+−++
∂
∂ II dgffgfffdu
t
h
ξφξφ
ξ
. (7.22)
Thus, if we approximate
u
by j
uand fby j
fwhich are polynomials of degree
1
−
K
, and if we set
)−−−+)−+= −+
ξξξξ
(])1()1([(])1()1([)()( LBupw,2/1RBupw,2/1 gffgfffF jjjjjj (7.23)
then, Eq. (7.22) can be written as
0])(
2
)([ =+
Ij
j
tj dF
h
u
ξφ
ξ
.(7.24)
Switching to the global coordinate,
0])()([ =+
j
Exjtj dxFu
φ
.(7.25)
Since both j
uand xj
F)( are polynomials of degree
1
−
K
, and the above holds for any polynomial
φ
of degree
1
−
K
,0)()( =+ xjtj Fu .(7.26)
In conclusion, the DG scheme is identical to the flux reconstruction scheme with the Radau polynomial as correction
functions. This completes the proof.

American Institute of Aeronautics and Astronautics
26
Note that the solution by DG method via (7.4) involves the inverse of the mass matrix due to the technique of
projecting onto the basis functions kj ,
φ
, which play the role of the test function. The solution via flux reconstruction
approach using Eqs. (7.26) and (7.23), however, involves no mass matrix regardless of the choice of solution points
since no test functions are employed.
Flux Reconstruction Schemes in DG Framework.
We now show the converse: for the case of advection, the flux reconstruction scheme can be cast in the DG
framework. The difference, compared with standard DG schemes, is the space of test functions: for DG, this space is
1−K
P; for the scheme defined by the flux correction function LB
gg =, in the case of advection equation
0=+ xt auu where 0
>
a, this space consists of polynomials
φ
of degree
K
such that
dx
d
/
φ
is orthogonal to
g
.
To proceed, let
g
be defined by 1)1(
=
−
g
and 0)1(
=
g
, and
g
be a polynomial of degree
K
on ]1,1[
−
=
I
(approximating 0 in some sense, but we do not need this property). Applying (7.23) and (7.26) to the case of
advection and using the local coordinate, we have )−−−+= −
ξξξ
(])1()1([)()( upw,2/1 gfffF jjjj ,(7.27)
and
0)()(
2
=+
ξ
jtj
jFu
h.(7.28)
To cast the above method in the DG framework, let
φ
be a polynomial of degree
K
on
I
. The above implies
0])()(
2
[=+
Ijtj
jdFu
h
ξφ
ξ
.(7.29)
Next, using (7.27),
[
]
{}
 )−−−+= −
Ijjj
IjdgfffdF
ξφξξξφ
ξ
ξ
()1()1()()( upw,2/1 .(7.30)
Separately, by integration by parts,
 −−−= Ijjj
Ijdfffdf
ξφξφφξφ
ξξ
)()1)(()1)(()( , (7.31)
Again, by integration by parts, since 1)1(
=
−
g
and 0)1(
=
g
,
 −(−1)−= II dgdg
ξφφξφ
ξξ
.(7.32)
Using the above two equations, (7.30) implies
[
]
[]


)−−−−
(−1)−−−−−−−=
−
−
Ijj
jj
Ijjj
Ij
dgff
ffdfffdF
ξφξ
φξφξφφξφ
ξ
ξξ
()1()1(
)1()1()()1)(()1)(()(
upw,2/1
upw,2/1 (7.33)
For the moment, assume that
φ
satisfies
0=
Idg
ξφ
ξ
. (7.34)
Observing that )1(
upw,2/1 jj ff =
+since the wind blows from the left, (7.33) then implies,
 −(−1)−−(1)= −+ Ijjj
IjdfffdF
ξφφφξφ
ξξ
)1()1()( upw,2/1upw,2/1 . (7.35)
Equation (7.29) and the above yield the following DG form for the solution:
[]
0
2upw =−+
∂
∂ ∂Ij
I
Ij
jdffdu
h
t
ξφφξφ
ξ
.(7.36)
This equation is identical to (7.4) except for the space of test functions denoted by
T
. This space consists of
polynomials
φ
of degree
K
such that, by (7.34),
ξ
φφ
=
′
is orthogonal to
g
.
To describe
T
, first we show that it is a subspace. Indeed, if
φ
and
ϕ
are in
T
, then it can easily be verified
that
βϕ
αφ
+
is also in
T
for any real number
α
and
β
.

American Institute of Aeronautics and Astronautics
27
(a) gK ~
,1−
Phas the same dimension as 2−K
P(b) The case
g
is the Radau polynomial
Figure 7.1. (a) gK ~
,1−
Pis of dimension
1
−
K
.Here, g
~
is the projection of
g
onto 1−K
P, and gK ~
,1−
Pis the
subspace consisting of all
ϕ
in 1−K
Pthat are orthogonal to g
~
.(b) If KR
Rg ,
=, then 2
~
,1 −− =KgK PP ; therefore,
1−
=K
PT .
Next, we show that
T
is of dimension
K
. To this end, recall that K
Pis of dimension
1
+
K
, and 1−K
P,
dimension
K
. Denote the projection of
g
onto 1−K
Pby g
~
. On 1−K
P, the collection of all
ϕ
that are orthogonal to
g
~
form a subspace denoted by gK ~
,1−
P, which is of dimension
1
−
K
. Since
φ
′
is in 1−K
Pand
g
is in K
P, the
condition that
φ
′
is orthogonal to
g
is equivalent to the condition that
φ
′
is orthogonal to g
~
, i.e.,
φ
′
lies in
gK ~
,1−
P. By observing that the derivative operator is a linear mapping from K
Ponto 1−K
P, which maps
T
onto
gK ~
,1−
P, and gK ~
,1−
Pis of dimension
1
−
K
, we conclude that
T
is of dimension
K
. This completes the proof.
VIII. Two Dimensional Extensions
For the two-dimensional (2D) extension of the current flux reconstruction approach, the solution procedure
reduces to a series of one-dimensional (1D) operations in a manner similar to the extension of the staggered-grid
scheme discussed in (Kopriva and Kolias 1996). Here, however, the schemes are simpler since they involve only
one grid instead of three.
With ),,( tyxuu =, the 2D conservation law takes the form
0=++ yxt gfu (8.1)
with initial condition ),()0,,( init yxuyxu =.(8.2)
We assume that
u
is periodic or of compact support so that boundary conditions are trivial. Denote
()
yx gfgf
yx
fgff +=








∂
∂
∂
∂
== ,.,)div(and),( rr .(8.3a, b)
Then the conservation law can be written as
0)div( =+ fut
r
.(8.4)
The simplest nontrivial two-dimensional conservation law is the advection equation: with constants
a
and b,
0=++ yxt ubuau .(8.5)
We will discuss the general case of unstructured quadrilateral meshes later. For the moment, let the domain of
calculation be divided into uniform cells; each cell is a rectangle ji
E,of length x
h, width y
h, and center
),( ,, jiji yx ,...,2,1,0,1...,
−
=
j
i
Next, in the local description, let the solution points on ]1,1[
−
=
I
be k
ξ
,
Kk ...,,1=. For simplicity, we use the same solution points in the
η
direction: ll
ξη
=,Kl ...,,1=. The
K
K
×
solution points on the biunit square 2
I
I
I
=
×
are given by the tensor product of the 1D solution points, namely,
),( lk
ηξ
where
K
k
...,,1
=
and
K
l
...,,1
=
. The solution points on ji
E,are defined by
)2/,2/(),( ,,,,,,,, yljixkjilkjilkji hyhxyx
ηξ
++= .
2−K
P
1
−
K
P
K
P
g
g
~
2−K
P
1
−
K
P
K
P
g
g
~
gK ~
,1−
P
2
~
,1 −−
=
KgK PP

American Institute of Aeronautics and Astronautics
28
On each rectangle ji
E,, let the solution
u
be approximated by
K
K
×
pieces of data lkji
u,,, at the solution points
),( ,,,,,, lkjilkji yx ,Klk ...,,1, =.
At time level n, suppose the data lkji
u,,, are known for all i, j, k, and l. We wish to calculate dttud lkji /)(
,,, at
time n
tt =, which is abbreviated to dtud lkji /
,,, , in terms of the data. In other words, we wish to evaluate x
fand
y
gat the solution points. Then, we march in time by, say, a Runge-Kutta method.
First, on ji
E,we construct a polynomial interpolating lkji
u,,, where kand lvary. Equivalently, in the local
coordinates ),
ηξ
( on 2
I
, we wish to construct a polynomial interpolating the data lkji
u,,, at the solution points
),( lk
ηξ
where kand lvary. To this end, with k
φ
denoting the basis function for the 1D case given in (2.7b), let
the basis functions on the biunit square 2
I
be defined via tensor product: for each fixed ),(
l
k
,Klk ...,,1, =,
)(((
,,
ηφξφηξφφ
lklklk )=),= .
Thus, lk,
φ
takes value 1 at the solution point ),( lk
ηξ
and 0 at all other
1
2
−
K
solution points. See Fig. 8.1. The
polynomial interpolating the data on the cell ji
E,is
 ==
))=),=), K
lk
lklkji
K
lk
lklkjiji uuu
1,
,,,
1,
,,,,, ((((
ηφξφηξφηξ
.(8.6)
Note that for each fixed
η
,),
ηξ
(
,ji
uis a polynomial of degree
1
−
K
in
ξ
, and vice versa. That is, ji
u,is in
1111 −−−− ⊗= KKK,K PPP ; e.g., for
2
=
K
,ji
u,is bilinear; for
4
=
K
, bicubic.
-1
-0.5
0
0.5
1
-0.5
0
0.5
1
1.5
(a)
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
0
1
-1
-0.5
0
0.5
(b)
Figure 8.1. Basis functions for 3
=
Kwith Gauss points as solution points. (a) The functions 2
φ
and 3
φ
for the
1D case. (b) The function )(( 2323
ηφξφφ
)=
,for the 2D case.
Let the solution point in the
η
direction be fixed, say 2
=
l, i.e., 2
=
ηη
. See Fig. 8.2(a). Along the
ξ
direction,
the
K
values 2,k
u, where Kk ≤≤1, determine a polynomial of degree
1
−
K
in
ξ
. Noting that 1)( 22 =
ηφ
and
0( 2=)
ηφ
lfor 2≠l, the above equation yields
)=))=),  ==
ξφηφξφηξ
(u((u(u
K
k
k,k,j,i
K
l,k
lkl,k,j,ij,i 12
122 .(8.7)
Thus, for each fixed l
η
, the interpolation along the
ξ
-direction reduces to a one-dimensional problem. (A similar
statement holds for each fixed k
ξ
, along the
η
-direction.) We can calculate the left and right cell edge values
),− 2, 1(
η
ji
uand ), 2, 1(
η
ji
uby the 1D procedure. Along each vertical edge of midpoint ),2/1(
j
i
+
(see Fig. 8.2(a)),
using the data in the adjacent cells, i.e., in ji
E,1−and ji
E,, at each l
η
, say 2
=
ηη
l, we can obtain the values just to
its left and right:
η
ξ
φ
ξ
φ

American Institute of Aeronautics and Astronautics
29
),−=),= +2,12, 1(and1(
ηη
jiRjiL uuuu .(8.8)
From these two values, we can obtain the upwind flux normal to the edge:
),(
upwupw RL uuff =.(8.9)
Next, we define the discontinuous flux functions. At each solution point, from lkji
u,,, , we can evaluate the
corresponding fluxes: )(and)( ,,,,,,,,,,,, lkjilkjilkjilkji ugguff == .(8.10)
On each cell ji
E,, the polynomial ji
f,interpolating the values lkji
f,,, is given by
 ==
))=),=), K
lk
lklkji
K
lk
lklkjiji fff
1, ,,,
1, ,,,,, ((((
ηφξφηξφηξ
.(8.11)
Similarly,
 ==
))=),=), K
lk
lklkji
K
lk
lklkjiji ggg
1, ,,,
1, ,,,,, ((((
ηφξφηξφηξ
.(8.12)
Again, let the solution point l
η
be fixed, say, 2
=
l. Along the
ξ
direction, the
K
values 2,k
f(where kvaries)
determine a polynomial of degree
1
−
K
in
ξ
)=), 
=
ξφηξ
((
12,,,2,
K
k
kkjiji ff .(8.13)
We can calculate the flux values at the left and right interfaces by ),− 2, 1(
η
ji
fand ), 2, 1(
η
ji
f.
(a) (b)
Figure 8.2. (a) Solution points (dots) and flux points on vertical boundaries (squares). At each of these flux
points, we define an upwind flux value upw
f. (b) Flux points on horizontal boundaries (squares).Here, we
define upw
g.
Next, to evaluate x
for equivalently (up to a constant factor)
ξ
ffor each fixed l
η
, say, 2
η
, we construct a
polynomial ), 2, (
ηξ
ji
Ffor the flux, which is of degree
K
in
ξ
and take the upwind flux values upw
fat the two
interfaces corresponding to
−1
=
ξ
and
1
=
ξ
. The procedure is exactly the same as that in Section 3. The
calculation of y
gis similar; we omit the details.
The solution procedure can be summarized as follows. First, calculate and store the
K
upwind (normal) fluxes
at each edge for all edges. Next, focusing on the cell ji
E,, evaluate and store lkji
f,,,
)(
ξ
and lkji
g,,,
)(
η
of the
discontinuous flux functions for all kand l. To correct for the left and right edges, for each fixed l, add the
correction terms to lkji
f,,,
)(
ξ
for
K
k
...,,1
=
as in the 1D case. In a similar manner, to correct for the bottom and top
edges, for each fixed k, add the correction terms to lkji
g,,,
)(
η
for
K
l
...,,1
=
. This completes the algorithm.
For the general case, we map the biunit square to a quadrilateral (with possibly curved edges) via an
isoparametric mapping. The problem then reduces to one similar to the above rectangular mesh case. The details are
again omitted.
Next, we carry out the Fourier (Von Neumann) analysis for the 2D case. Consider the advection equation
3
ξ
η
2
η
ξ
3
η
1
η
1
ξ
2
ξ

American Institute of Aeronautics and Astronautics
30
0=++ yxt ubuau .(8.14)
where )cos(
α
=
a
,)sin(
α
=
b
, and
α
is the flow angle. Note that the flow speed is 1. The cells employed for this
analysis are squares centered at ),(
j
i
:]2/1,2/1[]2/1,2/1[
,+−×+−= jjiiE ji . Due to symmetry, we only need to
consider
α
in ]4/,0[
π
. Thus, for the vertical edge identified by its midpoint ),2/1(
j
i
+
and the horizontal edge
)2/1,(
+
j
i
, the upwind values are the values from the cell ji
E,. Denote by ji ,
uthe column vector of 2
K
components )...,,...,,...,,,...,,,( ,,,1,,,,2,,1,2,,,1,,2,1,,1,1,,, KKjiKjiKjijiKjijijiji uuuuuuu=u.(8.15)
Similar to the 1D case, the solution can be expressed as
1,1,0,10,1,0,0
,−−−− ++= jijiji
ji
dt
duCuCuC
u.(8.16)
Here, 0,0
C,0,1−
C, and 1,0 −
Care 22
K
K
×
matrices. Let the wave number in the
x
-direction be x
wand that in the
y
-direction, y
w. Let the imaginary number be denoted by Iinstead of the standard notation ito avoid confusion
with the index i. Replacing ji ,1−
uby ji
wx
e,
u
I−and 1, −ji
uby ji
wy
e,
u
I−, we obtain
(
)
ji
w
w
ji y
xee
dt
d
,1,00,10,0
,uCCC
uI
I−
−
−
−++= .(8.17)
Therefore, the semidiscretization results in the equation
ji
ji
dt
d
,
,uS
u=(8.18)
For all schemes discussed here, Shas 2
K
eigenvalues. The eigenvalue which approximates the exact one is the
principal eigenvalue denoted by ),( yx wwS :
)(),( yxyx bwawiwwS +−≈ .(8.19)
All other eigenvalues are spurious ones. All eigenvalues must lie in the left half of the complex plane for the
semidiscretization to be stable.
-15-12.5 -10 -7.5 -5 -2.5
-10
-5
5
10
(a) DG
-10 -8 -6 -4 -2
-6
-4
-2
2
4
6
(b) SG
-6 -4 -2
-6
-4
-2
2
4
6
(c) Lump, Lo
-10 -8 -6 -4 -2
-6
-4
-2
2
4
6
(d) Gauss
-6 -4 -2
-6
-4
-2
2
4
6
(e) Lump, Ch-Lo
Figure 8.3. Spectra (collection of all eigenvalues) of 2D semidiscrete schemes, 3
=
K.Note that if the flow is
horizontal, then the solution reduces to that of the 1D case, and so does the spectrum. The spectrum for the case of
horizontal flow is shown by the small triangles; diagonal flow, small squares; and all flow directions, small dots.
For all tested cases, the boundary of the whole spectrum is 2times the spectrum of the 1D case (except when the
1D spectrum is not convex as in the case of Lo-ChLump,
gwith 8
=
K). The spectrum for the case of diagonal flow
also yields the boundary of the whole spectrum.

American Institute of Aeronautics and Astronautics
31
-25 -20 -15 -10 -5
-15
-10
-5
5
10
15
(a) DG
-15 -12.5 -10 -7.5 -5 -2.5
-10
-5
5
10
(b) SG
-12 -10 -8 -6 -4 -2
-10
-5
5
10
(c) Lump, Lo
-17.5-15-12.5-10-7.5 -5 -2.5
-10
-5
5
10
(d) Gauss
-20 -15 -10 -5
-10
-5
5
10
(e) Lump, Ch-Lo
Figure 8.4. Spectra of 2D semidiscrete schemes,
4
=
K
.
The spectra (collection of all eigenvalues) of the semidiscrete schemes are shown in Fig. 8.3 for the case 3
=
K,
and in Fig. 8.4,
4
=
K
. Note that if the flow is horizontal, then the solution reduces to that of the 1D case, and so
does the spectrum. The spectrum for the case of horizontal flow is shown by the small triangles; diagonal flow,
small squares; and all flow directions, small dots. For all tested cases, the boundary of the whole spectrum is 2
times the spectrum of the 1D case (except when the 1D spectrum is not convex as in the case of Lo-ChLump,
gwith
8
=
K). The spectrum for the case of diagonal flow also yields the boundary of the whole spectrum.
Next, we discuss accuracy. The orders of accuracy and errors of the 2D schemes for
4
=
K
are tabulated in
Table 8.1. Note that the order of accuracy for the 2D case is identical to that of the 1D case for all schemes. The
errors for schemes DG, Ga
g,LoLump,
g, and Lo-ChLump,
g(lumping for Chebyshev-Lobatto points), have negative
real parts; a closer check of the eigenvalues reveals that these four schemes are stable. The errors for the last three
schemes, namely, SG, 3
g, and Lo
ghave positive real parts; as a result, they are unstable, but the instabilityis mild.
(
4
=
K
)
Scheme in 2D Order of
accuracy Coarse mesh error, 7/
π
α
=
5/,3/
ππ
== yx ww
Fine mesh error, 7/
π
α
=
10/,6/
ππ
== yx ww
DG 7i
77 102.1109.8 −− ×−×− i
109 104.2106.3 −− ×−×−
Ga
g6i
66 107.8107.2 −− ×+×− i
88 102.7101.1 −− ×+×−
LoLump,2 gg =6i
56 105.1108.4 −− ×+×− i
78 103.110.2 −− ×+×−
Lo-ChLump,
g4i
44 107.3107.1 −− ×+×− i
56 102.1107.2 −− ×+×−
SG 4 i
55 106.7109.5 −− ×−× i
66 100.310.1 −− ×−×
3
g5i
44 107.1108.2 −− ×+× i
66 104.1100.5 −− ×+×
Lo
g4i
44 104.1100.1 −− ×−× i
66 102.5107.1 −− ×−×
Table 8.1. Errors and accuracy of 2D schemes for flow angle
7
/
=
π
α
and
4
=
K
.
IX. Numerical Results
As a preliminary test we apply the schemes discussed above to solve the advection equation. Periodic boundary
conditions and the standard 4-stage Runge-Kutta time stepping are employed for all problems. (Recall that the
correction function LoLump,
guses the Lobatto points as solution points and lumps the correction to the
corresponding boundary point; similarly, Lo-ChLump,
guses the Chebyshev-Lobatto points and lumps the correction
in the same manner; and the correction function Ga
gvanishes at the
1
−
K
Gauss points.) The solutions by the
schemes DG, SG, LoLump,
g,Ga
g,Lo-ChLump,
gand, for comparison, the standard piecewise-linear upwind scheme
(i.e., Van Leer’s MUSCL or Fromm’s scheme) are shown. For all plots, the crosses represent the exact solutions and
the dots, the numerical solutions.
The three tests involve the advection of a Gaussian wave. On the domain
]
1
,
0
[
, the initial condition is
])5.(40exp[)( 2
init −−= xxu .(9.1)
We employ 10 cells and, in each cell, four solution points. Thus, the wave is represented by a total of 40 data
points.

American Institute of Aeronautics and Astronautics
32
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(a) Piecewise-linear upwind
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(b) DG
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(c) Staggered-grid
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(a) LoLump,
g
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(b) Ga
g
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(c) Lo-ChLump,
g
Figure 9.1. Advection of a Gaussian wave.The wave travels a distance of 10 periods or 400 cells; the solution
points are the equidistant points (not including the two cell boundaries); the crosses represent the exact solutions
and the dots, the numerical solutions.
For the first test, equidistant (not including the cell boundaries) points are used as solution points. The wave
travels a distance of 10 periods, i.e., 400 cells. Each time step corresponds to a CFL number of approximately
.9*(CFL limit) in a manner that the distance of 10 periods is an integer multiple of the time step size. The numbers
of time steps are: for DG, 764; SG, 432; LoLump,
g, 385; Ga
g, 490; and Lo-ChLump,
g, 562. For the piecewise-linear
upwind scheme, a CFL number of .750469 is employed; after 533 time steps, the wave traveled a distance of 400
cells. The solutions are shown in Fig. 9.1. Note that the DG scheme yields the most accurate solution; on the other
hand, the piecewise-linear upwind and Lo-ChLump,
gschemes yield the least accurate ones.
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(a) Piecewise-linear upwind
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(b) DG
0.2 0.4 0.6 0.8 1 x
-0.2
0.2
0.4
0.6
0.8
1
y
(c) Staggered-grid
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(a) LoLump,
g
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(b) Ga
g
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(c) Lo-ChLump,
g
Figure 9.2. Advection of a Gaussian wave for a long distance.The wave travels a distance of 50 periods or
2000 cells; the solution points are the Gauss points. The DG scheme yields the most accurate solution; on the other
hand, the piecewise-linear upwind, staggered-grid, and Lo-ChLump,
gschemes yield the least accurate ones. Note that
the staggered-grid solution exhibits growing wiggles, which are a sign of instability which, in this case, is mild.

American Institute of Aeronautics and Astronautics
33
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(a) Piecewise-linear upwind
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(b) DG
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(c) Staggered-grid
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(a) LoLump,
g
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(b) Ga
g
0.2
0.4
0.6
0.8
1
x
0.2
0.4
0.6
0.8
1
y
(c) Lo-ChLump,
g
Figure 9.3. Advection of a Gaussian wave with an irregular mesh. The wave travels a distance of 10 periods or
400 cells. The mesh is irregular (random). The solution points are the Lobatto points.
The second test case is the same as the first except that the solution points are the Gauss points, and the wave
travels a distance of 50 periods, which corresponds to 2000 cells. For the piecewise-linear upwind scheme, a
uniform mesh is employed with a CFL number of .7499; after 2667 time steps, the wave travels a distance of 2000
cells. The solutions are shown in Fig. 9.2. Again, note that the most accurate solution is that of DG; among the least
accurate solutions are those of the piecewise-linear upwind and Lo-ChLump,
gschemes. Also note that the staggered-
grid solution exhibits growing wiggles, which are a sign of (mild) instability. Here, the solution is identical to that of
the original staggered-grid method.
The third test is also the same as the first except that the grids are generated randomly with the constraint that the
smallest cell is at least 1/5 the size of the largest. Here, except for the piecewise-linear upwind scheme, we use the
Lobatto points as solution points for all schemes. Note that the solutions are much less accurate than those of the
first test, which employ a uniform mesh. In general, the solutions are comparable to those in Fig. 9.2 where the wave
travels a distance five times longer.
X. Conclusions and Discussion
In summary, we introduced a new approach to high-order accuracy for the numerical solution of conservation
laws via the construction of a continuous flux function. This function is obtained by adding correction terms, which
account for the jumps in flux values at interfaces, to the discontinuous flux function. The approach bridges the two
existing approaches of DG and staggered-grid (or spectral difference) and leads to simplified versions of these two
methods. Schemes via the new approach are simple and economical: they solve the conservation laws in differential
form and involve only one grid, namely, the solution points; they are also conservative.
Among the new schemes (with
K
solution points), two are super-accurate of order
2
2
−
K
, one order less than
that of DG. The first is the scheme with the correction function LoLump,
gthat employs the Lobatto points as solution
points and lumps the correction to the corresponding boundary point. The second is the scheme with the correction
function Ga
g, which vanishes at the
1
−
K
Gauss points. Compared to the DG scheme, these two schemes have the
advantage that the corresponding CFL limits are not as restrictive: for LoLump,
g, it is roughly twice as large, and for
Ga
g, about three halves.
Concerning the property of super-accuracy or super-convergence, the following result was obtained via Fourier
(Von Neumann) analysis. With
K
solution points, if the correction function gis perpendicular to m
Pwhere
20
−
≤
≤
Km , then the scheme is accurate to order 1
+
+
mK .
The relation between the DG and new approaches was established: (1) the DG method can be cast via the new
approach, and (2) for an advection problem, schemes via the new approach can be cast in the DG framework. As a
result of the new approach, the DG scheme can be formulated using the differential instead of integral form.
Next, we discuss extensions and applications.

American Institute of Aeronautics and Astronautics
34
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Legendre Polynomials
and Gauss Points
P0
P1
P2
P3P4
Figure A.1. Legendre Polynomials and Gauss points.
Note the different scales in
ξ
and
η
axes.
1. The approach can be extended to the case of the Euler and Navier-Stokes equations on a 2D quadrilateral
mesh in a manner similar to the extension of the staggered-grid schemes (Kopriva 1998). The extension to the case
of a triangular mesh, however, remains to be explored. The key problem in the triangular mesh case is that the
concept of tensor product is not available.
2. The approach can be applied to solve ordinary differential equations (ODE). If we use the right Radau points
as solution points in the time direction and the right Radau polynomial as the correction function, the result is an
implicit Runge-Kutta scheme identical to the Radau IIA scheme (Hairer and Wanner 1991). The proof using our
approach is simpler than that of LeSaint and Raviart (1974) where a quadrature formula including the left boundary
was employed. Our approach is more direct: it employs a formula including the right boundary.
3. Using the application to ODE discussed above and some additional techniques introduced for the moment
scheme in (Huynh 2004) and employed in (Suzuki and Van Leer, 2007), one can derive a moment scheme of degree
K
for arbitrary
K
, which is explicit and has a CFL limit of1 for the case of one spatial dimension. Such a scheme
can be considered as an extension of Van Leer’s scheme 3 (1977), which is a piecewise linear scheme for advection.
It can also be considered as an explicit space-time DG scheme with the property that the CFL limit is 1, whereas the
CFL limit for the combination of a method of line (or Runge-Kutta) and DG is considerably smaller.
4. Concerning the recent spectral difference and spectral volume schemes on a triangular mesh, two questions
can be raised. (1) Are these schemes stable or mildly unstable? (2) Can we derive spectral schemes that share the
properties of stability and super-accuracy of the scheme Ga
gdiscussed above?
Appendix: Polynomials, Quadratures, and Related Topics
Since the Legendre, Radau, and Lobatto polynomials play a key role in our approach, they are reviewed below.
We also introduce new polynomials and derive additional properties needed.
A. Legendre Polynomials
For any smooth functions
v
and
w
on the interval ]1,1[
−
=
I
, let the 2
L
inner product be defined by
1
1− ))=
ξξξ
dwvwv ((),( . (A.1)
Let
m
be a nonnegative integer, and let m
Pbe the
space of polynomials of degree
m
or less. Then
m
Pis a vector space of dimension 1
+
m. The
1
+
mmonomials m
ξξ
...,,,1 form a basis for m
P,
which is not orthogonal. Using the Gram-Schmidt
process, we can derive an orthogonal basis from a
nonorthogonal one. The result is the Legrendre
polynomials k
P, ...,2,1,0=kwhere k
Pis scaled
so that 1)1( =
k
P. The Legendre polynomial of
degree kis given by the following recurrence
formula (see, e.g., Szego 1967 or Hildebrand 1974):
,1,0 01 ==
−PP
and, for 1≥k,
)
−
−)
−
=) −−
ξξξξ
(
1
(
12
(21 kkk P
k
k
P
k
k
P.(A.2)
From their definitions, the Legendre polynomials satisfy, for lk ≠,
0((),( =))= 1
1−
ξξξ
dPPPP lklk . (A.3)
For kl =, we have the following property, which is useful in the standard formulation of DG scheme, but is not
employed here,
12
2
)((),( 2
+
=)== 1
1− k
dPPP kkkk
ξξγ
. (A.4)
ξ
η

American Institute of Aeronautics and Astronautics
35
The first few Legendre polynomials are
)33035(),35(),13(,,1 24
8
1
4
3
2
1
3
2
2
1
210 +−=−=−===
ξξξξξξ
PPPPP .
The plots of these polynomials are shown in Fig. A.1. Note that 1)1( =
k
Pfor all k, while 1)1( =−
k
Pfor even k,
and 1)1( −=−
k
Pfor odd k; in addition, k
Pis an even function (involves only even powers of
ξ
) for even k, and
an odd function for odd k.
Next, for nonnegative integers kand
m
, if mk >, then the Legendre polynomial k
Pis perpendicular to m
P,
the space of polynomials of degree
m
or less. In particular, 1+m
Pis perpendicular to m
P, i.e., for any polynomial
p
of degree
m
or less,
0((),( 11 =))= 1
1− ++
ξξξ
dpPpP mm . (A.5)
Readers who are not interested in the proofs can skip the rest of this subsection with no loss of continuity.
We will need the values )1(−
′
k
Pand )1(
′
k
P, which can be calculated as follows. First, rewriting Eq. (A.2) with
kreplaced by 1+k, we have
)
+
−)
+
+
=) −+
ξξξξ
(
1
(
1
12
(11 kkk P
k
k
P
k
k
P. (A.6)
Differentiating the above,
)
′
+
−)
′
+)
+
+
=)
′−+
ξξξξξ
(
1
](([
1
12
(11 kkkk P
k
k
PP
k
k
P.
Evaluating at
1
=
ξ
, we obtain
)
′
+
−
′
+
+
+
+
+
=)
′−+ 1(
1
)1(
1
12
1
12
1( 11 kkk P
k
k
P
k
k
k
k
P.
Subtracting )1(
′
k
Pfrom both sides,
)
′
+
−
′
+
+
+
+
=
′
−)
′−+ 1(
1
)1(
1
1
12
)1(1( 11 kkkk P
k
k
P
k
k
k
k
PP .
Multiplying by 1+k,
]1()1([12])1(1([)1( 11 )
′
−
′
++=
′
−)
′
+−+ kkkk PPkkPPk .
Denoting )1()1( 1
′
−
′
=−kkk PPD , the above implies
12)1( 1++=+ +kDkDk kk .
As a result, kDk=.
By the definition of k
D, therefore,
kPP kk +)
′
=
′
−1()1( 1.
Since 1
0=P,0)1(
0=
′
P. The above implies, for 0>k,
)1(...21)1( 2
1+=+++=
′
kkkPk.(A.7)
As for the left boundary, since k
Pis even for even k, and odd for odd k,)1()1()1( 1′
−=−
′−k
k
kPP , i.e.,
)1(
2
)1(
)1( 1+
−
=−
′−kkP k
k.(A.8)
Next, the following two relations (see (8.5.7) of Hildebrand 1974) between
′
k
Pand k
Pwill be employed later (in
deriving the key property of LoLump,
g): for 1≥k,
)+)−=)
′
−−
2
ξξξξξ
((()1( 1kkk PkPkP ,(A.9)
and
)+−)+=)
′
−+
2
ξξξξξ
()1(()1(()1( 1kkk PkPkP .(A.10)
Consider the right hand sides of (A.9) and (A.10). The term )
ξξ
(
k
Pdrops out if we multiply (A.9) by 1+kand
(A.10) by k, and then add the results,

American Institute of Aeronautics and Astronautics
36
](([)1(()1)(1( 11 )−)+−=)
′
−++ −+
2
ξξξξ
kkk PPkkPkk .(A.11)
That is, for 1≥k,
)
′
−
+
++
−=−) 2
−+
ξξξξ
()1(
)1(
)1(
)(( 11 kkk P
kk
kk
PP .(A.12)
B. Gauss Quadratures
Let
1
≥
K
be an integer and fa smooth function on ]1,1[
−
=
I
. The integral 1
1− )
ξξ
df (can be approximated
using
K
evaluations of fby

=
1
1− ≈) K
k
kk fwdf
1
)((
ξξξ
;(A.13)
here, both the weights k
wand the sample points k
ξ
are parameters, which are to be determined. Therefore, the total
number of unknowns is
K
2
. Setting k
f
ξ
=, 12...,,1,0 −= Kk , respectively, and requiring the above
approximation to be exact results in a system of
K
2
equations with
K
2
unknowns. This system determines k
w
and k
ξ
, for Kk ...,,1=. The resulting formula =
K
kkk fw
1)(
ξ
is called a Gauss quadrature. The values k
ξ
,
Kk ...,,1=are the Gauss (quadrature) points.
With just
K
evaluations, the Gauss quadrature has a degree of precision of
1
2
−
K
, i.e., for any polynomial
p
of degree
1
2
−
K
or less, the Gauss quadrature for
p
is exact:

=
1
1− =) K
k
kk pwdp
1
)((
ξξξ
;
In general, a quadrature rule with
K
evaluations has a degree of precision no higher than
1
−
K
. For Gauss
quadratures, however, with two Gauss points (
2
=
K
), the degree of precision is 3 (exact for cubics); with three
Gauss points, the degree of precision is 5.
We are now ready to show that the Gauss points K
ξξ
...,,
1defined by the quadrature (A.13) are identical to the
K
zeros of the Legrendre polynomial K
P. Consider the polynomial
q
of degree
K
, which vanishes at the
K
Gauss points, and takes value 1 at
1
=
ξ
:
∏
=−
−
=) K
kk
k
q
11
(
ξ
ξξ
ξ
.(A.14)
We wish to prove that
q
is identical to K
P. Since the Legendre polynomial K
Pis a polynomial of degree
K
determined by the conditions that it is orthogonal to 1−K
Pand 1)1( =
K
P, we need to show that
q
satisfies these
conditions. By the definition of
q
above, we have 1)1( =q. Next, let
r
be an arbitrary polynomial of degree
1
−
K
.
Then
qr
is of degree
1
2
−
K
and vanishes at the
K
Gauss points. Since the Gauss quadrature has degree of
precision
1
2
−
K
, we conclude that 0=
1
1−
ξ
dqr . As a result,
q
is orthogonal to 1−K
P. This completes the proof.
For
1
=
K
, the Gauss point is 0, and for
K
up to 4, the Gauss points are
2
=
K
577350.0−=
1
ξ
3/1577350.0
2==
ξ
3=K774597.0−=
1
ξ
0
2=
ξ
774597.0
3=
ξ
4
=
K
861136.0−=
1
ξ
339981.0
2−=
ξ
339981.0
3=
ξ
861136.0
4=
ξ
For a fixed
K
, the weights k
w's of the Gauss quadrature with
K
evaluations at the Gauss points k
ξ
are given by
2
1
2
2
2
2)]([)1(
)1(2
)]()[1(
2
kK
k
kKk
kPK
P
w
ξ
ξ
ξξ
+
+
−
=
′
−
=.(A.15)
For the case of four Gauss points, the weights are

American Institute of Aeronautics and Astronautics
37
347855.0=
1
w652145.0
2=w652145.0
3=w.347855.0
4=w
C. Radau Quadratures and Polynomials
Again, let
1
≥
K
be an integer and fa smooth function on ]1,1[
−
=
I
. Consider a quadrature formula that
includes one of the two boundary points, say, the right boundary
1
=
ξ
:
1where)((
1
=≈) 
=
1
1− K
K
k
kk fwdf
ξξξξ
,(A.16)
All
1
−
K
other sample points k
ξ
are parameters (unknowns); all k
w,Kk ...,,1=are also parameters. The total
number of parameters is therefore
1
2
−
K
. To solve for these unknowns, we need
1
2
−
K
equations, which can be
obtained by setting k
f
ξ
=, 22...,,1,0 −= Kk , respectively, in the above. This system of equations yields all k
w’s
and
1
−
K
values k
ξ
, 1...,,1 −= Kk . These k
ξ
together with 1=
K
ξ
are called the right Radau points; the
corresponding quadrature, the right Radau quadrature. The left Radau points and quadrature are defined in the same
manner with the right boundary point replaced by left. Clearly, the left Radau points are reflections (mirror images)
of the right Radau points about 0.
With
K
evaluations (including that at 1=
K
ξ
), the Radau quadrature has a degree of precision of
2
2
−
K
(as
opposed to
1
2
−
K
of the Gauss quadrature), i.e., it is exact for any polynomial of degree
2
2
−
K
or less.
While the Gauss points are the zeros of the Legendre polynomials, the Radau points are the zeros of the Radau
polynomials discussed below. First, for any fixed positive integer
m
, denote by m
Pr the projection from the space
of polynomials onto m
P, i.e., for any polynomial
p
, the polynomial )(Pr p
mis the least-squares approximation for
p
on
I
by a polynomial of degree
m
. Clearly, if
p
is of degree
m
or less, then pp
m=)(Pr .
We can now define the Radau polynomials. To avoid the factor K
)1(−, we deal with the left Radau polynomials.
Consider the Legendre polynomials K
Pand 1−K
P. By definition, they are perpendicular to 2−K
P, i.e.,
0)(Pr)(Pr 122 == −−− KKKK PP . Next, let
p
given by
1−
+= KK PPp
βα
where
α
and
β
are parameters to be determined. Clearly,
p
is perpendicular to 2−K
P. If we require
p
to satisfy
1)1(and0)1( ==− pp ,
then,
α
and
β
are determined. Indeed, since 1)1( =−
k
Pfor even k, and 1)1( −=−
k
Pfor odd k(see Fig. A.1),
while 1)1( =
k
Pfor all k, we have 2/1==
βα
. The resulting
p
is the left Radau polynomial of degree
K
denoted by KL
R,:
)( 1
2
1
,−
+= KKKL PPR ,(A.17)
1)1(and0)1( ,, ==− KLKL RR .(A.18)
Here the letter
R
stands for ‘Radau’ and subscript
L
for ‘left’. Thus, KL
R,is a polynomial of degree
K
determined uniquely by the two conditions of (A.18) and
1
−
K
conditions that it is perpendicular to 2−K
P.
We can now show that the left Radau points k
ξ
,Kk ...,,1=(where 1−=
1
ξ
) defined by a quadrature formula
similar to (A.16) are identical to the
K
zeros of KL
R,. Indeed, consider the polynomial
q
of degree
K
, which
vanishes at the
K
left Radau points, and takes value 1 at the right boundary 1
=
ξ
:)
ξ
(qis given by expression
(A.14) with the Gauss points replaced by the Radau points. We wish to show that
q
is identical to KL
R,. Since
KL
R,is uniquely determined by the conditions that it is orthogonal to 2−K
Pand 0)1(
,=−
KL
Rand 1)1(
,=
KL
R, it
suffices to show that
q
satisfies these conditions. By its definition,
q
satisfies that latter two conditions. Next, let
r
be an arbitrary polynomial of degree
2
−
K
. Then
qr
is of degree
2
2
−
K
and vanishes at the Radau points.
Therefore, due to the degree of precision of
2
2
−
K
of the Radau quadrature, we have 0=
1
1−
ξ
dqr . That is,
q
is
orthogonal to 2−K
P. This completes the proof.

American Institute of Aeronautics and Astronautics
38
For a fixed
K
, the weights k
w's of the left Radau quadrature with
K
evaluations at the left Radau points k
ξ
are given by, for Kk ...,,1=,







=
≠
′
−
=
−
=−
−
.1if
2
,1if
)]([
2
1
1
)]([
11
2
2
1
2
1
2
k
K
k
P
PK
wkK
k
kK
k
k
ξ
ξ
ξ
ξ
(A.19)
As for the right Radau polynomials of degree
K
denoted by KR
R,, which vanishes at the right boundary,
)(
2
)1( 1, −
−
−
=KK
K
KR PPR ,(A.20)
0)1(and1)1( ,, ==− KRKR RR .(A.21)
For a fixed
K
, the weights k
w's of the right Radau quadrature with
K
evaluations at k
ξ
are given by







=
≠
′
+
=
+
=−
−
.if
2
,if
)]([
2
1
1
)]([
11
2
2
1
2
1
2
Kk
K
Kk
P
PK
wkK
k
kK
k
k
ξ
ξ
ξ
ξ
(A.22)
We will also need the slopes of the Radau polynomials at the boundary points. Since for any integer 0>k,
=
−
−=−
′k
l
k
klP 1
1
)1()1( , by expression (A.17) for KL
R,,





−+−=−
′
+−
′
=−
′ −
=
−
=
−
−
1
1
2
1
1
1, )1()1(
2
1
)]1()1([
2
1
)1( K
l
K
K
l
K
KKKL llPPR .
-1
-0.5
0
0.5
1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Radau Polynomials
RL,1
RL,2
RL,3
RL,4
(a)
-1
-0.5
0
0.5
1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Radau Polynomials
RR, 1
RR, 2
RR, 3
RR, 4
(b)
Figure A.2. Radau Polynomials on ]1,1[
−
=
I
. (a) Left; (b) Right.
Noting that 1
)1( −
−Kand 2
)1( −
−Kare of opposite sign, the quantity in the above square bracket is K
K1
)1( −
−. Thus,
for any integer
1
≥
K
,
KR K
KL
2
)1(
)1( 1
,
−
−
=−
′.(A.23)
At the boundary
1
=
ξ
, by (A.17) and (A.7),
[]
)1()1(
2
1
)]1()1([
2
1
)1( 1, −++=
′
+
′
=
′−KKKKPPR KKKL .
Or, for any integer
2
)1( 2
,K
RKL =
′.(A.24)
D. Lobatto Quadratures and Polynomials
ξ
ξ

American Institute of Aeronautics and Astronautics
39
Let
2
≥
K
be an integer and fa smooth function on ]1,1[
−
=
I
. Consider a quadrature formula that includes
both boundary points:
1and1where)(( 1
1
=−=≈) 
=
1
1− K
K
k
kk fwdf
ξξξξξ
,(A.25)
while all other k
ξ
are parameters (unknowns); all k
w,Kk ...,,1=are also parameters. The total number of
unknowns is therefore
2
2
−
K
. To solve for these unknowns, we need
2
2
−
K
equations, which can be obtained by
setting k
f
ξ
=, 32...,,1,0 −= Kk , respectively, in the above. This system of equations yields all k
w’s and
2
−
K
values k
ξ
where 1...,,2 −= Kk . These k
ξ
together with 1
1−=
ξ
and 1=
K
ξ
are called the Lobatto points; the
corresponding quadrature, Lobatto quadrature.
With
K
evaluations (including those at the two boundaries), the Lobatto quadrature has a degree of precision of
32
−
K(as opposed to
2
2
−
K
of the Radau and
1
2
−
K
of the Gauss quadratures), i.e., it is exact for any
polynomial of degree 32
−
Kor less.
Next, we derive the Lobatto polynomials. Consider the Legendre polynomials K
Pand 2−K
P. They are
perpendicular to 3−K
P, i.e., 0)(Pr)(Pr 233 == −−− KKKK PP . Set
2
Lo −
−= KKK PP .(A.26)
Then K
Lo is also perpendicular to 3−K
P. In addition,
0)1(Lo)1(Lo ==− KK .(A.27)
The zeros of the above Lobatto polynomial are the Lobatto points. Denoting them by k
ξ
,Kk ...,,1=where 1
1−=
ξ
and 1=
K
ξ
, then the weights k
w's for the corresponding quadrature are







==
−
<<
−
=−
.or1if
)1(
2
,1if
)]([
1
)1(
2
2
1
Kkk
KK
Kk
P
KK
wkK
k
ξ
(A.28)
-1
-0.5
0
0.5
1
-1.5
-1
-0.5
0
0.5
1
Lobatto Polynomials
Lo5
Lo4
Lo3
Lo2
(a)
-1
-0.5
0
0.5
1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Radau Polynomials
and Lobatto Points
RL,3
Rwa
RL,4
(b)
Figure A.3 (a) Lobatto Polynomials on ]1,1[
−
=
I
. (b) Polynomial wa
Rand the Radau Polynomials. The
polynomial wa
R reaches extrema at the Lobatto points (big dots) except at the right boundary; it is the correction
function LoLump,
gfor the right boundary. Note the different scales in
ξ
and
η
axes.
The Lobatto polynomial can also be written as
′
−
−
−
+
−
=) −
21
)1(
)1(
)]1([
(Lo KK P
KK
KK
ξξ
.(A.29)
Moreover, it can be expressed in terms of the Radau polynomials: for
2
≥
K
,
)()1(2)(2Lo 1,,1,, −− −−=−= KLKL
K
KLKLK RRRR .(A.30)
ξ
η
ξ
η

American Institute of Aeronautics and Astronautics
40
Similar to the case of the Gauss and Radau points, the Lobatto points defined by the quadrature formula (A.25)
are identical to the zeros of the Lobatto polynomials defined by (A.26). They are also given by
ξ
-coordinates of the
intersections of the graphs of KL
R,and 1, −KL
Ras shown in Fig. A.3(b) (or those of KR
R,and 1, −KR
R).
E. Radau Polynomials, Lobatto Points, and Lumping.
Let
2
≥
K
be an integer. To avoid carrying the factor K
)1(−, consider the left Radau polynomials KL
R,and
1, −KL
R. Their graphs intersect at
K
points whose
ξ
-coordinates are the Lobatto points, which are the big dots in
Fig. A.3 (b). Let wa
Rbe the weighted average
1,,wa )1( −
−+= KLKL RaaRR (A.31)
where
a
is chosen so that
−1
=
ξ
is a double zero of wa
R, i.e., 0)1(
wa =−
′
R. To calculate
a
, by (A.23),
)1(
2
)1(
)1(and
2
)1(
)1( 2
1,
1
,−
−
=−
′
−
=−
′−
−
−KRKR K
KL
K
KL .(A.32)
The above two slopes are of opposite sign; as a result, for )1(
wa −
′
Rto be 0,
1
2
1
−
−
=
K
K
a.(A.33)
That is,
1,,wa
1
2
1
2
1−
−
+
−
−
=KLKL R
K
K
R
K
K
R.(A.34)
Note that whereas 0)1(
wa =−
′
R, at the other boundary, a straightforward calculation using (A.24) yields
)12(2
)1()1(
)1(
12
)1(
12
1
)1( 22
1,,wa −
−+−
=
′
−
+
′
−
−
=
′−K
KKKK
R
K
K
R
K
K
RKLKL .
That is,
)1(
2
1
)1(
wa −=
′KKR .(A.35)
Note that this quantity is exactly the inverse of the Lobatto weight at the boundaries given by Eq. (A.28).
We claim that wa
Rhas extrema at all
K
Lobatto points except the right boundary. To prove this claim, it
suffices to show that
′
wa
Rvanishes at all Lobatto points except
1
=
ξ
. Since wa
Ris a polynomial of degree
K
,
′
)− wa
1( R
ξ
is also of degree
K
. It suffices then to show that
′
)− wa
1( R
ξ
vanishes at all Lobatto points. By
definition (A.26) for the Lobatto polynomials, the problem reduces to showing that
)(const)1( 2wa −
−=
′
−KK PPR
ξ
.(A.36)
where ‘const’ stands for some constant. Or, equivalently,
)((1const)1( 2wa −
2−)+=
′
−KK PPR
ξξ
.(A.37)
To show that the above equation holds, first, expand wa
Rin terms of the Legendre polynomials: by (A.34) and
(A.17),
.])12()1([
)12(2
1
)(
2
1
12
)(
2
1
12
1
21
211wa
−−
−−−
+−+−
−
=
+






−
++






−
−
=
KKK
KKKK
PPKPK
K
PP
K
K
PP
K
K
R
(A.38)
Next, denote by
Q
the quantity in the above square bracket:
21
)12()1( −− +−+−= KKK PKPKPKQ .(A.39)
Then, by differentiating
Q
and then multiplying by )1( 2
−
ξ
, we have
′
−+
′
−−+
′
−−=
′
−−
2
−
222 21 )1()1()12()1()1()1( KKK PKPKPKQ
ξξξξ
.(A.40)
We now rewrite the above in terms of k
Pin two different expressions. For the first expression, by applying (A.9) to
the first term on the right hand side of (A.40) and (A.10) to the last two, we obtain

American Institute of Aeronautics and Astronautics
41
[
]
[
]
[]
.)1()1(
)12()1()1(
12
11
−−
−−
2
−−−+
−−++−−=
′
−
KK
KKKK
PKPKK
PKPKKPKPKKQ
ξ
ξξξ
That is, after rearranging and observing that the term 1
)1( −
−K
PKK drops out,
{
}
.)1()12()12()1()1( 21 −−
2−+−+−−−−=
′
−KKKK PKPKPKPKKQ
ξξξξ
(A.41)
For the second expression, by applying (A.9) to the first two terms on the right hand side of (A.40) and (A.10) to the
last term, we obtain
[
]
[
]
[]
.)1()1(
)1()1()12()1()1(
12
211
−−
−−−
2
−−−+
−+−−−++−−=
′
−
KK
KKKK
PKPKK
PKPKKPKPKKQ
ξ
ξξξ
That is, after rearranging and observing again that the term 1
)1( −
−K
PKK drops out,
{
}
.)12()12()1()1( 221 −−−
2+−+−−−−=
′
−KKKK PKPKPKPKKQ
ξξξξ
(A.42)
By adding the quantities in the curly brackets in (A.42) and (A.41), the term 1
)12( −
−K
PK
ξ
drops out. Thus,
2
)1()12()1()12()1(
1
11 −
2+−++−−=
′
−






−
+KK PKPKQ
KK
ξξξ
(A.43)
The above simplifies to
)()1(
)1(
12−
−−=
′
−
−KK PPQ
KK
ξ
(A.44)
Equations (A.38) and (A.39) imply ])12(2[/
wa −= KQR . By differentiating and then multiplying by )−
ξ
1( , we
have ])12(2[/1(1( wa −
′
)−=
′
)− KQR
ξξ
. The above then implies
)(
)12(2
)1(
1( 2wa −
−
−
−
−=
′
)− KK PP
K
KK
R
ξ
(A.45)
Therefore, wa
Rhas extrema at all
K
Lobatto points except the right boundary. This completes the proof.
Acknowledgments
The author wishes to thank Dr. Kopriva for several fruitful discussions. He is also grateful to Drs. Ananda
Himansu, Ali Ameri, and David L. Rigby for their reviews and comments.
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