Content uploaded by H.T. Huynh
Author content
All content in this area was uploaded by H.T. Huynh on Feb 13, 2017
Content may be subject to copyright.
A Survey of the Isentropic Euler Vortex Problem using
High-Order Methods
Seth C. Spiegel,∗H. T. Huynh,†and James R. DeBonis‡
NASA Glenn Research Center, Cleveland, OH, 44135, USA
The flux reconstruction (FR) method offers a simple, efficient, and easy to implement
method, and it has been shown to equate to a differential approach to discontinuous
Galerkin (DG) methods. The FR method is also accurate to an arbitrary order and the
isentropic Euler vortex problem is used here to empirically verify this claim. This problem
is widely used in computational fluid dynamics (CFD) to verify the accuracy of a given
numerical method due to its simplicity and known exact solution at any given time. While
verifying our FR solver, multiple obstacles emerged that prevented us from achieving the
expected order of accuracy over short and long amounts of simulation time. It was found
that these complications stemmed from a few overlooked details in the original problem
definition combined with the FR and DG methods achieving high-accuracy with minimal
dissipation. This paper is intended to consolidate the many versions of the vortex problem
found in the literature and to highlight some of the consequences if these overlooked details
remain neglected.
I. Introduction
Accurate and efficient numerical methods for the simulation of complex aerodynamic flows are desired
to help reduce the design and development costs of aerodynamic vehicles and aeropropulsion systems. The
primary challenge lies in the prediction of the turbulent motion within the flow and its effect on the overall
motion of the fluid. Advanced methods such as large-eddy simulation (LES) and direct numerical simulation
(DNS) offer great advancements in predicting turbulence, but these methods require significant improvements
in accuracy and efficiency in order to become widely used. Focusing on either the accuracy or efficiency aspect
of the underlying numerical algorithms, when developing a computational fluid dynamics (CFD) code, usually
demands concessions from the other parameter.
Promising results have been found applying LES in the realm of aeropropulsion, but the spatial and tem-
poral resolution needed for LES requires high-order (higher than second) numerical schemes historically only
found in structured grid methods with large stencils. These methods have numerous drawbacks: difficulty in
grid generation, high sensitivity to grid quality, complex boundary conditions, and poor scalability for paral-
lel computing. These drawbacks severely limit the ability of these codes to compute complex aeropropulsion
configurations, e.g., noise suppressing nozzles.1
In an attempt to alleviate the difficulties of structured grid generation for complex geometries, researchers
have applied LES methods to unstructured grids.2,3However, the limited second-order accuracy of current
unstructured methods requires an exorbitant number of grid points and offers little incentive over the tra-
ditional structured methods.4,5The flux reconstruction (FR) method6–8provides a simple, efficient, and
easy to implement method for solving the Navier-Stokes equations on unstructured grids and is accurate to
an arbitrary order. By employing the high-order FR method with LES on unstructured grids, limitations
related to structured grid generation, grid resolution, and computational efficiency can be overcome.9–12
∗NASA Postdoctoral Fellow, Inlets and Nozzles Branch, 21000 Brookpark Road, and AIAA Member.
†Aerospace Engineer, Inlets and Nozzles Branch, 21000 Brookpark Road, and AIAA Member.
‡Aerospace Engineer, Inlets and Nozzles Branch, 21000 Brookpark Road, and AIAA Associate Fellow.
1of21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
22nd AIAA Computational Fluid Dynamics Conference
22-26 June 2015, Dallas, TX
AIAA 2015-2444
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
AIAA AVIATION Forum
II. Numerical Methods
II.A. Spatial Discretization
We briefly review the FR method below. Consider the 1D conservation law
ut+fx= 0 (1)
where the subscripts tand xdenote partial differentiation with respect to time and space, respectively.
Let the computational domain Ωbe divided into an arbitrary number of nonoverlapping cells. Each cell is
transformed from physical space within the global domain to a common reference space/element, I, which
in 1D is the line interval, I= [−1,1]. With rdenoting the local coordinate variable in the reference space,
the mapping function r=χ(x) is used to complete this transformation. Similarly, the inverse function
x=χ−1(r) maps the reference element back to physical space. The details of such mappings, especially for
higher dimensions, are beyond the scope of this paper; the interested reader can find additional information in
many texts on spectral-/finite-element type methods, including but not limited to Ref. 13–16. The following
review focuses on using the FR method to solve the transformed version of equation (1) within each reference
element.
At a given time t, let the exact solution to the transformed conservation law within the reference element
be given by ue(r, t) which may be a nonsmooth and/or discontinuous function. Let u(r, t) be a polynomial
of degree Pand approximate ue(r, t) within the reference element. This approximate solution polynomial,
u(r, t), is local to each element, and is generally discontinuous across cell interfaces. With the dependence
on time being understood, we will drop the variable tfrom ue(r, t) and u(r, t) to simplify notation.
Let NPrefer to the minimum number of solution points needed to define the solution polynomial within
the reference element, and let ξdenote the set of solution points, ξn,n= 1,2, . . . , NP. Using tensor products
to extend this to higher dimensions, the N-dimensional conservation law requires NP= (P+ 1)Nsolution
points within each computational cell.
The Lagrange polynomial is defined as the interpolating polynomial through a set of nodal points, for
example ξ, that takes the value 1 at ξnand zero at all other points ξi6=n:
`n(r) =
NP
Y
i=1
i6=n
r−ξi
ξn−ξi
(2)
Note that the Lagrange polynomial in equation (2) is a polynomial of degree P=NP−1 and the set of
Lagrange polynomials for all solution points in ξis a set of NPpolynomials each of degree P.
Anodal coefficient of the solution, denoted by ˜un, is simply the value of the approximate solution
polynomial evaluated at the solution point, ξn, i.e., ˜un=u(ξn). Let ˜
uPdenote a vector containing the set
of NPnodal coefficients for the solution at each of the solution points. Assuming that all of ˜
uPis known,
we can use the Kronecker delta property of equation (2) to construct the approximate solution polynomial
as
u(r) =
NP
X
n=1
˜un`n(r) (3)
The above relation is why the set of Lagrange polynomials can also be referred to as a nodal basis set. The
tilde accent is used in this section to identify the nodal coefficients for a polynomial function.
The derivative of the solution polynomial is a linear combination of the derivatives of the polynomial
basis functions
d
dr u(r) = ur(r) =
NP
X
n=1
˜und
dr `n(r)(4)
Using the previous equation, we can define the derivative operator, Dr, as the NP×NPmatrix with elements
Dr[i, j] = d
dr `j(r)
ξi
(5)
that transforms the nodal coefficients, ˜
uP, into the derivative nodal coefficients, ˜
ur. Note that this derivative
is exact only when the function being differentiated is a polynomial of degree Por less.
2of21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
If we use the nodal solution to evaluate the flux function, f, at each solution point, then interpolate
these NPvalues by a polynomial of degree P, we get what is called the discontinuous flux function. It only
uses information inside a given cell and does not account for interaction between it and the data in the
neighboring cells. Using the superscript ‘D’ for discontinuous, the discontinuous flux polynomial, fd
(r), is
given by
fd
(r) =
NP
X
n=1
˜
fn`n(r) (6)
where the nodal coefficients of the discontinuous flux, ˜
fn, are evaluated at the solution points using the
nodal coefficients of the solution, i.e., ˜
fn=f(˜un). This collocation projection results in fd
(r) being a
polynomial of degree P, which only matches the true flux function if the conservation law in equation (1) is
the linear advection equation. For a nonlinear conservation law such as those found in the Euler equations
or Navier-Stokes equations, the true flux function evaluated using the solution polynomial is of much higher
degree than Pand possibly even irrational, and fd
(r) above may not provide a good approximation. We
will explore other ways of approximating the true flux function later.
With fd
(r) given by equation (6), we can compute the derivative of the approximated discontinuous flux
using the derivative operator from equation (5). Denoting ˜
fas the column vector containing the coefficients
˜
fn,n= 1,2, . . . , NP, the derivative d
dr [fd
(r)] can be approximated using the matrix-vector product
˜
fr=Dr˜
f(7)
where, ˜
fris the column vector containing the values of the approximate flux derivative at each solution point.
Note that the collocation projection of the discontinuous flux in equation (6) is of degree P. Consequently,
its derivative is of degree P−1.
If we employ the above fd
rto evaluate fxfor the conservation law, we obtain erroneous solutions since
such a derivative does not include the interaction of the data between adjacent cells. Therefore, we seek to
construct a so-called continuous flux function, denoted by F(r), which accounts for this interaction between
adjacent cells and approximates the discontinuous flux function in some sense, to which we can then calculate
its derivative. The continuous flux function will be obtained by adding a correction to fd
, the discontinuous
one.
To this end, at each (cell) interface, let uland urbe the values taken on by the solution polynomials
to its left and right, respectively. An upwind flux value common for the two adjacent cells, denoted by
f∗=f∗(ul, ur), can then be defined via the wind direction. Here, for the case of the Euler equations, we
use Roe’s flux17 with an entropy fix.18
Let f∗
lb denote the upwind flux at the ‘left boundary’ of the cell and f∗
rb denote the upwind flux at the
‘right boundary’ of the cell. We can then construct the continuous flux function, F(r), as a polynomial of
degree P+1 by requiring that it takes on the upwind flux values at the two cell interfaces, and it approximates
fdvia P−1 conditions. Instead of defining F, we define F−fd
, which approximates the zero function. At
the two interfaces r=±1, the function F(r)−fd
(r) takes on the following values, which are called the flux
jumps:
F(−1) −fd
(−1) = f∗
lb −fd
(−1) and F(1) −fd
(1) = f∗
rb −fd
(1) (8)
Therefore, F(r)−fd
(r) has prescribed left and right interface values, is of degree P+ 1, and approximates
zero.
We now separate the prescription of the jump at the left interface from that of the right. This separation
plays a critical role in the new approach. Again, using the local coordinate, let glb be the correction function
for the left interface of the reference element defined by
glb(−1) = 1, glb (1) = 0 (9)
and glb is a polynomial of degree P+ 1 approximating the zero function in some sense. We always choose
grb by reflection: grb(r) = glb (−r). As a result,
grb(−1) = 0, grb (1) = 1 (10)
Consider the left interface r=−1. The polynomial
fd
(r)+[f∗
lb −fd
(−1)] glb(r) (11)
3of21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
provides a correction at the left interface for fd
(r) by changing the flux value at this interface from fd
(−1)
to f∗
lb, while leaving the value at the right interface unchanged, namely, fd
(1). Next, the polynomial
F(r) = fd
(r)+[f∗
lb −fd
(−1)] glb(r)+[f∗
rb −fd
(1)] grb(r) (12)
provides corrections to both interfaces; using equations (9) and (10), one can easily verify that F(−1) = f∗
lb
and F(1) = f∗
rb. Thus, the above F(r) is of degree P+ 1, takes on the upwind flux values at the two
interfaces, and approximates fd
(r) in the same sense that glb and grb approximate the zero function.
The derivative of F(r) at the solution point ξnis evaluated as
∂ F (r)
∂r
ξn
=Fr(ξn) = ˜
fr,n + [f∗
lb −fd
(−1)] g0
lb(ξn)+[f∗
rb −fd
(1)] g0
rb(ξn) (13)
Finally, Fris transformed from reference space to physical space using the inverse mapping χ−1, where it
can then be employed to approximate fxat each solution point within the physical grid cell. The solution
to equation (1) is then advanced in time via an explicit Runge-Kutta method.
In order to complete this brief review of the FR method, we must define the correction function glb; the
definition of grb follows by reflection as previously discussed. With Lkdenoting the (standard) Legendre
polynomial of degree k, the right Radau polynomial of degree k(with k > 0) is defined by
RR,k =(−1)k
2(Lk−Lk−1) (14)
A family of correction functions for the left boundary that results in stable schemes is given by
glb =α RR,P +1 + (1 −α)RR,P with 0 ≤α≤1 (15)
Note that α= 1 results in the discontinuous Galerkin (DG) method, and α=P+1
2P+1 results in the modified
Spectral Difference (SD) method that is stable.
II.B. Temporal Discretization
The results presented in this paper were integrated in time using the 3-stage strong stability preserving (SSP)
Runge-Kutta method,19 which is 3rd-order accurate in time. All of the results used a constant (global) time
step to advance the solution in time. The size of the time step for each simulation was chosen to be sufficiently
small meaning further reduction in its size would not produce significant changes in the solution error.
III. Governing Equations
The Euler equations describe the motion of an inviscid fluid such as the isentropic vortex that is the
topic of this paper. The 2D Euler equations for a calorically perfect, compressible fluid can be written in
differential form as
∂
∂t
ρ
ρvx
ρvy
ρE
+∂
∂x
ρvx
ρv2
x
ρvxvy
vx(ρE +p)
+∂
∂y
ρvy
ρvxvy
ρv2
y
vy(ρE +p)
= 0 (16)
where ρ,vi,p, and Erespectively refer to the density, velocity in coordinate direction i, pressure, and the
total energy per unit mass of the system. Thermodynamic closure is found through the total energy equation
ρE =p
γ−1+1
2ρv2
x+v2
y(17)
where γis the ratio of specific heats for the fluid in question. Additionally, the ideal gas law
p=ρRgasT(18)
is utilized to relate the temperature, T, to the pressure and density where Rgas is the specific gas constant
of the fluid. The speed of sound, a, is computed using the relation
a=rγp
ρ=pγRg asT(19)
4of21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
In the present work, the fluid is assumed to be calorically perfect air where γ= 1.4 and the dimensionalized
specific gas constant is Rgas = 287.15J/(kg·K).
IV. Isentropic Euler Vortex Problem
IV.A. Problem Definition
The isentropic Euler vortex problem is commonly used9,14,20–30 for testing the order of accuracy of a numeri-
cal method because it is easy to implement and the exact solution is known at all times. This problem is ideal
for high-order methods because their theoretical accuracy should allow them to propagate the vortex with
minimal entropy production for an indefinite amount of time. Being able to sustain vortical flow structures
without the numerical method contributing unwanted numerical dissipation is crucial for advanced turbu-
lence methods like LES. All of these reasons are why this is becoming a popular test case for high-order
methods that are designed to produce minimal numerical dissipation.
There are numerous variations in the definition of the vortex problem throughout literature. To the
authors’ knowledge, the first version of this problem was used by Shu20 to compare high-order weighted
essentially non-oscillatory (WENO) finite-volume (FV) methods with a traditional second-order FV method.
The definition for the vortex problem that is presented here modifies this first version in an attempt to unify
many of the variations that exist.
IV.A.1. Analytic Solution
The initial conditions for the vortex problem are found by superimposing perturbations in velocity and
temperature onto a uniform mean flow. These perturbations are defined such that the entropy, S=p/ργ,
is constant throughout the entire grid domain, i.e., δS = 0. By compounding various parameters, these
perturbations are ultimately based upon a Gaussian function of the form
Ω = β ef(20)
where βis the maximum strength of the perturbation and the function fdetermines the strength of the
perturbation at a particular grid coordinate through the relation
f(x, y) = −1
2σ2x
R2
+y
R2(21)
The parameter σis the familiar standard deviation term found in a general Gaussian function that is used
to control the spread of the distribution, whereas Ris a characteristic length scale for the grid. These two
parameters, σand R, could have been combined into a single parameter to further simplify the problem
definition, but they have been kept separate in order to unify dimensional and nondimensional versions of
the vortex problem. Keeping these separate will also prove to be useful later when looking at what effects
the spread of the perturbation can have on the results of a simulation.
Using the Gaussian function in equation (20), the perturbations in velocity and temperature are given
by
δvx=−y
RΩ
δvy= + x
RΩ
δT =−(γ−1)
2σ2Ω2
(22)
These perturbations, along with the isentropic relations between density, pressure, and temperature, are
used to define the initial flow conditions of the primitive variables as
˜ρ0= (1 + δT )
1
γ−1
˜vx,0=M∞cos α+δvx
˜vy,0=M∞sin α+δvy
˜p0=1
γ(1 + δT )
γ
γ−1
(23)
5of21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
Table 1: Parameters defining the isentropic Euler vortex problem. Quantities for these parameters are
given for a few versions of this problem that can be found in literature. All values without units are
nondimensional.
α M∞ρ∞p∞T∞Rgas R σ β L
Shu20 45°q2
γ1 1 1 – 1 1 M∞5√2
4πe
1
25
Vincent24 90°0.4 1 1
γM 2
∞1 – 3
21M∞27
4πe
2
920
Hesthaven and Warburton14 0°q1
γ1 1 1 – 1 q1
2M∞5
2πe15
High-Order Workshop (Slow)25 0°0.05 – 100 kPa 300 K 287.15 J
kg·K
1
200 11
50
1
20
High-Order Workshop (Fast)25 0°0.5 – 100 kPa 300 K 287.15 J
kg·K
1
200 11
5
1
20
Stationary Vortex 0°0 1 1 1 – 1 1 5
2π√γe
1
210
Above, Mrepresents the Mach number with the subscript ∞indicating a mean flow quantity, the parameter
αis the angle of attack for the mean flow which determines the direction of propagation for the vortex,
and the subscript 0 denotes a quantity given at t= 0. The tilde accents on the primitive variables indicate
that each is a nondimensional quantity where ρ∞,a∞, and T∞have been used as the characteristic density,
velocity, and temperature scales, respectively.
To further simplify the problem, an assumption has been made in equations (21) and (22) that the center
of the vortex lies at the coordinate origin. This assumption will continue for the remainder of this work where
the computational domain is defined to be a square that is centered about the origin. Some versions of this
problem use a grid domain where the coordinate origin is located on one of the grid boundaries invalidating
this assumption. A simple fix can be found by simply replacing the coordinate variables xand ywith ¯xand
¯ywhere
¯x=x−xc
¯y=y−yc
(24)
and the coordinate pair (xc, yc) gives the location of the vortex center.
Nine total parameters must be provided in order to use the above definition for the vortex problem. Five
of these parameters are standard requirements to define any general inviscid flow: α,M∞, and any three of
ρ∞,p∞,T∞, and Rgas. Three of the remaining four parameters required, R,σ, and β, control the shape of
the Gaussian function that is perturbing the mean flow. Finally, the parameter Lis equal to half the length
of the computational domain and will be discussed further in the following sections. Given the proper set of
these parameters to use with equations (20) to (23), many of the variations to the isentropic Euler vortex
problem can recovered. A few examples of these variations are shown in table 1which illustrates the range
in differences between some of the formulations.
IV.A.2. Computational Domain
The last remaining item needed in order to complete the problem is the definition of the computational do-
main. Two items pertaining to the computational domain have been mentioned thus far: the computational
domain is square and the center of the vortex is located at the coordinate origin. This implies that the
boundaries of the grid are located at plus/minus some length Lin both the xand ydirections. A general
guideline for determining the value for Lwill be discussed in section IV.B.1.
Finally, we need to define the boundary conditions to completely define the computational domain. The
boundary conditions used vary greatly between the different versions of this problem. Examples include but
are not limited to: periodic in all directions, periodic in one direction and imposed free-stream conditions in
the other direction, imposed analytical solution on the boundaries, and characteristic boundary conditions.
Periodic boundary conditions are by far the most popular and are required in order to model the vortex for
any significant amount of time. Analytical boundary conditions will not be explored in this work because
they prevent a true measure of the error generated by a numerical method. Each time the vortex passes
across a grid boundary, the vortex profile is essentially reset to the initial conditions removing all accumulated
error.
6of21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
X
Y
-5 0 5
-5
0
5
Density
0.98
0.92
0.86
0.8
0.74
0.68
0.62
0.56
0.5
Figure 1: Contours of the initial density profile for the Shu version of the vortex problem.
To illustrate the end product of this problem definition, a contour plot showing the initial density profile
for the Shu vortex is found in figure 1. The original version of this problem used a computation domain
of L= 5 and all boundaries were set to periodic conditions. Each simulation was run for a total of 10
time periods, where a period refers to the amount of time it takes for the center of the vortex to propagate
completely through the computational domain and return back to its initial location.
IV.B. Propagating Vortex
IV.B.1. Verification of Numerical Accuracy
In the literature, the primary use of this problem is to verify the numerical accuracy of a method using
grid refinement (h-refinement) and this is where we will begin as well. This problem is ideal for evaluating
the numerical accuracy of a method because the exact solution to the problem is just the translation of the
initial conditions based on propagation speed and simulation time. Therefore, at any given time, we can
easily compute the error between the simulation result and the translated initial conditions.
We started by running the Shu vortex problem on Cartesian grids containing cell counts of 202, 402, 602,
802, and 1002. For each grid that was used, we ran simulations for solution polynomials of degree one (P1)
through six (P6) amounting to a total of 30 individual simulations. Each simulation was run through one
period and a log-log plot of the L2error versus length scale was created. The length scale, h, is defined as
h=1
√NDOF
(25)
where NDOF is the total number of solution points (degrees-of-freedom) used for a simulation. Figure 2
shows two of these plots; the left plot shows the density error versus length scale whereas the right plot
shows the x-velocity error versus length scale.
The density error converges as expected except for the P6 solution on the finer grid levels. However,
something is clearly limiting the lower bound of the velocity error for the P3–P6 solutions. After further
testing, it was determined that the exponential function within the velocity perturbations (equations (20)
to (22)) is not zero at the domain boundaries. This fact combined with the periodic boundaries creates
two very small shear layers, one in each coordinate direction. An illustration of how these shear layers are
artificially created by the periodic boundaries is shown in figure 3. The errors associated with these shear
layers are orders of magnitude larger than the error from the numerical method resulting in the error limit
seen in the right plot of figure 2.
The fix to this problem is to make sure that the velocity perturbations are essentially zero at the grid
boundaries. From the problem definition given in section IV.A.1, the three parameters L,R, and σmust
satisfy the relation
ε≥L
Rexp −L2
2R2σ2(26)
7of21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
XX
X
X
X
XX
X
X
X
X
Log10 h
Log10 ( L2 Error )
-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6
-12 -12
-10 -10
-8 -8
-6 -6
-4 -4
-2 -2
P1
P2
P3
P4
P5
P6
X
X
X
X
X
X
X-Velocity Error
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Log10 h
Log10 ( L2 Error )
-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6
-12 -12
-10 -10
-8 -8
-6 -6
-4 -4
-2 -2
P1
P2
P3
P4
P5
P6
X
X
X
X
X
X
Density Error
Figure 2: L2errors for the Shu vortex problem after one period using a domain of size [−5,5] ×[−5,5].
Coupled Vortex from
Periodic Boundaries
Simulated Vortex
(a) Vortex coupling due to periodic boundary conditions.
X
Y
-5 0 5
-5
0
5
Error ||Velocity||
2.5E-05
2.2E-05
1.9E-05
1.6E-05
1.3E-05
1E-05
7E-06
4E-06
1E-06
(b) Velocity magnitude errors after 0.912 periods.
Figure 3: Errors due to the artificial shear layers that are created because the grid domain is not large
enough for the perturbations to reach zero at the domain boundaries.
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Log10 h
Log10 ( L2 Error )
-3.2 -3 -2.8 -2.6 -2.4 -2.2 -2
-12 -12
-10 -10
-8 -8
-6 -6
-4 -4
-2 -2
P2
P3
P4
P5
P6
X
X
X
X
X
Density Error
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Log10 h
Log10 ( L2 Error )
-3.2 -3 -2.8 -2.6 -2.4 -2.2 -2
-12 -12
-10 -10
-8 -8
-6 -6
-4 -4
-2 -2
P2
P3
P4
P5
P6
X
X
X
X
X
X-Velocity Error
Figure 4: L2errors for the Shu vortex problem after one period using a domain of size [−10,10] ×[−10,10].
8of21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
where εis some small number very near zero. Since all of our computations are performed using double
precision (64-bit) real numbers, we chose εto be 64-bit machine epsilon, ε64 = 2−52; this is the smallest
positive number capable of being represented by a 64-bit real number. Using ε64 and keeping the values for
Rand σconstant, solving equation (26) for Lindicates that Lmust be greater than 8.8. If machine epsilon
for quadruple precision is used, this increases even further to 12.7
In order to eliminate the artificial shear layers, Lwas increased from 5 to 10 and the previous simulations
were redone. Since the length of the domain in each direction was doubled, we also doubled the number of
cells in each direction to maintain the same grid cell sizes. This resulted in the cell counts for the five grid
levels becoming 402, 802, 1202, 1602, and 2002. The solution errors after one period using the new grids are
shown in figure 4. This shows that the size of the grid domain needs to be increased to perform an accurate
h-refinement test using this problem as originally defined in Ref. 20.
IV.B.2. Long Term Stability
The motivation for this section originated from trying to replicate the results found in Ref. 23 regarding the
stability of various nodal quadratures within triangular elements. Their work used a P4 solution polynomial
and the same problem definition as Shu except that they increased the size of their computational domain
to L= 10. The grid used for their study was an unstructured grid with 800 triangular cells formed by
diagonalizing a regular Cartesian grid with 202cells. They found that using the α-optimized nodal points14
based on the 1D Legendre-Gauss-Lobatto points produced large aliasing instabilities, eventually causing their
simulation to blow up at t= 360. A second simulation was performed using a nodal set from Ref. 31 that
provides a better quadrature rule within the reference triangle. The resulting “simulation was not afflicted
by aliasing instabilities and remained stable until it was stopped at t= 4000”.23
To best replicate these results, we made sure that the same problem definition, computational grid, and
numerical methods identified within their work were used. We found that using the α-optimized points
caused our code to blow up at t= 359.38, reproducing the results of their first simulation. Figure 5shows
the density contours at t= 358 and illustrates the instabilities within the vortex just before the simulation
finally blows up. This figure is similar to figure 12 in Ref. 23.
Instead of using the same quadrature rule for the second simulation, barycentric coordinates were used to
map the 1D Legendre-Gauss-Lobatto nodes to the reference triangle. Surprisingly, this very simple mapping
was significantly more stable than the α-optimized points for this problem with the simulation eventually
blowing up just before t= 900 or 45 periods. When examining the results for this simulation, we found
an odd trend in that the error appeared to increase exponentially through approximately 15 periods before
slowing to a linear accumulation for the remainder of the simulation. These trends are shown along with the
error at each period in figure 6.
These trends in the error accumulation contradict all previous experience with numerical methods where
the error accumulates linearly before it starts to diverge at an exponential rate. In trying to understand this
phenomenon, we switched to the use of quadrilateral cells over triangles. This eliminated any uncertainty
in the stability of the quadrature rules for triangles replacing it with proven quadrature rules based on
a tensor-product formulation of the standard 1D Legendre-Gauss or Legendre-Gauss-Lobatto quadratures.
Another concern was that this grid was too coarse for the version of the problem being used because the
majority of the vortex perturbations were contained within only a 62section of grid cells. We therefore chose
the 402, 802, and 1602sized grids with P2–P4 solutions to see what effects h/p-refinement has on the long
term stability for this problem. The log of the L2error in density versus the number of periods transversed
is shown for each of these cases in figure 7.
Each of these plots can be broken down into three distinct regions. The first region, beginning at t= 0,
sees an exponential increase in error and generally lasts for the first three to five periods. When the solution
is initialized, the analytical solution is evaluated at each solution point within the grid to provide the exact
initial conditions. This initialization is essentially a collocated projection of the analytical solution onto the
solution polynomial space. This exponential increase in error over the first few periods is caused by the error
associated with projecting the initial conditions.
After a few periods, the second region begins with the error leveling out and it then starts to increase
at a linear rate. This region represents the error accumulation of the numerical method once the initial
collocation projection has relaxed to its best approximation of the exact solution in the solution polynomial
space. Ideally, we want whatever method we are using to sustain this linear accumulation of error indefinitely.
9of21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
X
Y
-4 -2 0 2 4
-4
-2
0
2
4
Density
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
Figure 5: Contours of density at t= 358 using the
α-optimized nodal points. This figure can be used
to compare the presented results to those found in
Ref. 23 since this simulation attempted to verify
our code by replicating their results.
Period
Log10 ( L2 Error )
0 10 20 30 40 50
-3.5 -3.5
-3 -3
-2.5 -2.5
-2 -2
-1.5 -1.5
-1 -1
Density Error at Each Period
Exponential Fit: Periods 1-16
Linear Fit: Periods 17-44
Figure 6: L2norm of density error at each sim-
ulation period with overlaying trend lines. The
green trend line identifies the initial exponential
accumulation of error whereas the blue trend line
shows the subsequent linear accumulation before
the simulation eventually blows up.
Period
Log10 ( L2 Error )
0 20 40 60 80 100
-7 -7
-6 -6
-5 -5
-4 -4
-3 -3
-2 -2
-1 -1
P2
P3
P4
402 Grid
Period
Log10 ( L2 Error )
0 20 40 60 80 100
-8 -8
-7 -7
-6 -6
-5 -5
-4 -4
-3 -3
-2 -2
-1 -1
P2
P3
P4
802 Grid
Period
Log10 ( L2 Error )
0 20 40 60 80 100
-10 -10
-9 -9
-8 -8
-7 -7
-6 -6
-5 -5
-4 -4
-3 -3
-2 -2
P2
P3
P4
1602 Grid
Figure 7: L2norm of errors for the Shu vortex problem through 100 periods.
Unfortunately, all of these simulations contain a third region that begins after approximately 20 periods
and consists of another exponential accumulation of error. This time the error accumulates until it starts
destabilizing the vortex along with the simulation itself. What before appeared to be a contradiction of past
experience with numerical methods was actually what we should be expecting with a diverging solution.
However, the error levels off again once it reaches the range 0.01–0.001, almost as if the simulation has
re-stabilized itself. Clearly there is some kind of instability present within all of these simulations that is
causing them to diverge.
It is nearly impossible to understand what is causing these instabilities just by looking at how the total
error within the system evolves over time. We can achieve a greater understanding of this phenomenon by
looking at how the solution itself is evolving. Figure 8presents snap shots of the density contours at different
points in time for a simulation using a P3 solution on a grid with 1602cells. Below the time lapse is a plot
of the L2norm of the density error with green lines indicating the times for each of the snap shots. After
20 periods, the error begins to coalesce in the outer regions of the domain which coincides with the start of
the third region where the error accumulates at an exponential rate. Now we can see that this accumulation
is simply the error continuing to coalesce in the outer regions. After approximately 60 periods, the error
outside the vortex appears to reach a point of maximum saturation and the total density error levels off.
This error that has accumulated in the outer regions rotates around the vortex slowly deforming it more and
more as the simulation continues over the next 90–100 periods. Around 150 periods, the strain on the vortex
10 of 21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
caused by the surrounding error finally reaches a critical point, and it quickly gets torn apart. This point
of “collapse” is generally where a simulation will blow up due to a negative density or pressure. However,
sometimes the simulation will continue on for an indefinite amount of time with the remnants of the vortex
aimlessly floating around within the computational domain.
We tried various strategies to eliminate this instability such as reducing the time step size, increasing
the computational domain to L= 20 as well as L= 40, and even running the simulation using quadruple
(128-bit) precision. We also tried changing the explicit Runge-Kutta method used for time integration,
specifically the 5-stage/4th-order SSP32 and 5-stage/4th-order low-storage33 variations. Furthermore, we
also tried changing the upwinding method used to compute the common interface fluxes, specifically trying
the HLLC (Harten, Lax, and van Leer Riemann-solver with restored contact surface),34 AUSM+ (improved
Advection Upstream Splitting Method),35 and LDFSS (Low-Diffusion Flux-Splitting Scheme)36 upwinding
methods.
After all of these attempts failed, we wanted to determine if these instabilities were due to the nonlinearity
of the Euler equations. By altering the problem definition so that the velocity perturbations are zero and the
pressure is held constant, the momentum and energy equations reduce down to the continuity equation or
more simply the linear advection equation. These changes result in the simulation of a density/entropy wave
in the form of a Gaussian distribution that is convected at a constant velocity. Using this linear advection
problem, we were able to run this problem successfully for 800 periods using a P3 solution, Lobatto points,
and a 202grid with L= 5. The error in figure 9shows that the error accumulation is linear for all 800
periods with no signs of it diverging at any time soon afterwards.
These results appeared to indicate that we were experiencing the infamous aliasing instabilities that are
inherent to DG methods when applied to nonlinear equations. Common methods for suppressing aliasing
error are filtering the higher-order modes of the solution polynomial14,37 and increasing the quadrature
within each cell which is sometimes referred to as over-integration.37–39 Application of an exponential
filter14 to this problem failed at preventing the error from accumulating at an exponential rate regardless
of the parameters used for the filter function. A more extensive study using over-integration is found in
Ref. 40. In that study, we found that over-integrating was successful in eliminating aliasing errors caused by
using an insufficient quadrature to integrate the nonlinear flux functions. However, over-integration was not
successful in preventing the vortex problem from diverging for the simulations shown in figure 7. In fact,
the simulation shown in figure 8was over-integrated to 2Pyet still succumbed to this instability.
We performed one final sanity check by independently verifying our results using the finite-difference code
WRLES41 (Wave Resolving Large-Eddy Simulation). WRLES uses dispersion-relation-preserving (DRP)
schemes to solve the Favre filtered Navier-Stokes equations, and is accurate up to 12th-order in space. The
solution found by WRLES for this version of the vortex problem was very similar to the FR results and
exhibited the same trends in the accumulation of error.
Similar results by both codes is an indication this nonlinear instability is independent of the numerical
method being used, and the instability is more likely caused by the definition of the problem itself. Most of
the versions to this problem use either periodic or analytic boundary conditions, allowing for the vortex to
propagate indefinitely. However, Vincent et al. made a subtle new change to the boundary conditions of the
problem in Ref. 24, without providing a reasoning for the change. As others had done before, the free-stream
velocity was changed so that the vortex propagated in only one direction, in this case the y-direction only,
i.e. vx,∞= 0. The new twist was that the constant xboundaries were set to free-stream conditions instead
of leaving them periodic. With this slight change, they were able to successfully perform an h-refinement
study using the vortex problem and showed super accuracy through 45 periods without any instabilities.
We made a similar change to our problem by propagating the vortex in only the x-direction, i.e. vy,∞= 0,
and changing the constant yboundaries to characteristic boundary conditions. A comparison between be-
tween the version using all periodic boundaries and the version using a mixture of periodic and characteristic
boundary conditions is shown in figure 10. The simulation using a mixture of boundary conditions remains
stable through 200 periods, whereas the simulation with full periodic boundary conditions starts experi-
encing instabilities after only 20 periods. Clearly we have determined that the nonlinear instability we are
encountering is related to the periodicity of the boundary conditions.
IV.C. Stationary Vortex
We have successfully determined that the source of the nonlinear instability we have been encountering is
the periodicity of the boundary conditions. A subtle consequence of using periodic boundaries is that the
11 of 21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
Figure 8: Time series showing how the
density solution evolves over time when
all domain boundaries are periodic. The
plot to the right shows the L2norm of
the density error versus period. The
embedded green lines correspond to the
times at which each of the above contour
snapshots were taken.
Period
Log10 ( L2 Error )
0 50 100 150 200
-8 -8
-7 -7
-6 -6
-5 -5
-4 -4
-3 -3
-2 -2
-1 -1
12 of 21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
Period
Log10 ( L2 Error )
0 200 400 600 8 00
-5 -5
-4.5 -4.5
-4 -4
-3.5 -3.5
-3 -3
-2.5 -2.5
-2 -2
-1.5 -1.5
-1 -1
Euler Vortex
Advection Density Wave
Figure 9: L2norm of density error comparing the
accumulation of the error over the number of peri-
ods between the linear advection equation and the
Euler vortex problem.
Period
Log10 ( L2 Error )
0 50 100 150 200
-8 -8
-7 -7
-6 -6
-5 -5
-4 -4
-3 -3
-2 -2
All Periodic B.C. with
Diagonal Propagation
Constant-x Boundaries Periodic,
Constant-y Boundaries Characteristic,
and Propagation in x-direction
Figure 10: Differences in density errors over 200
periods due to boundary conditions.
problem we are solving is no longer the exact problem that was intended by the original definition. As we
showed in figure 3a, the periodic boundaries actually create a coupled vortex across each of the domain
boundaries. So instead of simulating a single, uncoupled, propagating vortex as intended by the original
problem definition, we are actually simulating an infinite array of vortices with centers that are separated by
a distance of 2×L. Even if the computational domain is made very large, eventually there will be interaction
between the coupled vortices due to the nonlinearity of the Euler equations.
To further understand the effects the boundary conditions have on this problem, we will change the
problem so the velocity of the mean flow is zero. This creates a stationary vortex in that the center of
the vortex never moves from its initial location throughout the simulation. This removes all dependence on
periodic boundaries since the vortex is no longer required to propagate across the domain boundary. This
is also the only way to simulate a single, decoupled vortex for a large length of time without having to use
moving/deforming mesh methods or an excessively large domain.
Because the vortex is no longer propagating, a small complication arises in that we no longer have a
definition for the amount of time in one period. In order to keep comparisons simple, we will define one
period for the stationary vortex as the same amount of nondimensional time required for the propagating
vortex to complete one period.
In the following two subsections, we present results for this stationary version of the problem using two
different sets of boundary conditions. The first case is the same as the original propagating version with all
periodic boundaries and the second case sets all boundaries to characteristic boundary conditions.
IV.C.1. Periodic Boundary Conditions
The first test case presented for the periodic stationary vortex is a P3 solution on a grid containing 1602
cells. Additionally, a P6 quadrature was used to over-integrate the solution in an effort to prevent aliasing
instabilities. A time lapse showing the density error propagating throughout the computation domain is
given in figure 11. An interesting observation is that the density error emanates as waves outward from the
center of vortex. For this particular problem, these error waves reach the boundary around t= 6. After
this point in time, the periodic boundaries cause the waves to be recirculated back into the domain from
the opposite boundary. Because the numerical method is continuously generating error within the vortex
interior, the recirculating error waves begin interacting with each other. These errors start building over time
and eventually they reach a large enough magnitude that causes the vortex to become unstable, resulting in
the solution diverging.
A parametric study was performed using P3–P6 solutions on grids containing 402, 802, and 1602cells
13 of 21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
X
X
X
X
X
X
XXXXXXX
X
X
XXXXXXXXXXXXX
X
X
XXXXXXXXXXXXXX
XXXXXXXXXXXXXXXX
Period
Log10 ( L2 Error )
0 20 40 60 80 100
-12 -12
-10 -10
-8 -8
-6 -6
-4 -4
-2 -2
P3
P4
P5
P6
40x40
80x80
160x160
X
Exponential Curve Fit
Figure 12: Density errors for the stationary vortex with all periodic boundaries.
so that we could get a sense of how the periodic stationary vortex is affected by h/p-refinement. The error
for each of these cases through 100 periods is shown in figure 12. We see the same results as before with
the error showing the same three regions described previously in section IV.B.2. A very noteworthy fact of
figure 12 is that the third regions of nearly all cases are now aligned. Specifically, only two of the twelve are
not aligned: the P3 solution on the 402grid and the P6 solution on the 1602grid. For the P3–402solution,
the error due to the grid resolution and numerical method is large enough that the vortex becomes unstable
before it is able to reach the area where the error accumulates at an exponential rate. The P6 solution
has the exact opposite problem. The combination of the high-order polynomial solution combined with the
higher fidelity grid results in a larger sensitivity to machine precision type errors. Running the P6–1602case
with decreasing values for the size of the time step created large horizontal shifts in both directions for the
start of the third region. Essentially these should both be treated as anomalies and separated from the other
aligned cases.
Added to figure 12 is a diagonal cyan line that was created from an exponential fit of the data in the
third region. The equation defining this line is
y=2.22 ×10−16e0.3x(27)
The remarkable part of this curve fit is the value of the coefficient in front of the exponential term, which is
the same as machine epsilon for a 64-bit floating point number. This appears to indicate that this exponential
error accumulation is due to machine rounding errors related to the floating point precision. It appears that
this machine error accumulates behind the scenes, masked by the discretization error of the problem.
IV.C.2. Characteristic Boundary Conditions
Because the vortex is no longer propagating, there is no longer a dependence on the domain boundaries
being periodic. This allows us to set all of the boundaries to characteristic boundary conditions allowing
for information to flow in/out of the domain as needed. The initial test case used in section IV.C.1 that
produced figure 11 was performed a second time with the only change being that all boundary conditions
are now characteristic boundaries. An identical time lapse to that in figure 11 but with all characteristic
boundary conditions is shown in figure 13. Here, the waves of error that emanate from the vortex leave
15 of 21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
the computational domain permanently once they reach the domain boundaries, thus preventing interaction
with older, recirculating waves of error.
Now that we have removed this instability, we can perform an h-refinement study using the vortex problem
over a large number of periods. To complete this study, we used P2–P4 solutions on grids containing 402,
802, 1202, 1602, and 2002cells. The density error versus period time for all but one of these cases is shown
in figure 14. Unfortunately, we were unable to prevent the P2 simulation on the 402grid from diverging due
to insufficient resolution, thus those results have been omitted.
To see how the order of accuracy develops over time, a linear fit was computed at each point in time
using the density error from each of the grid levels. The results of this linear fit using all of the simulations
in figure 14 produces the plot shown in figure 15a. If we examine the accuracy after a single period, the
order of accuracy appears to be P+ 1 as expected with values of 3.3, 4.8, and 5.5 for the P2, P3, and P4
solutions, respectively. However, if we were to do the same analysis after five periods, the respective values
change to 4.0, 5.8, and 5.5 indicating that the P3 solution is of higher order than the P4 solution. Jumping
all the way to 100 periods, it appears that the P2 solution is 6th order accurate and the P3 solution is nearly
8th order accurate.
These odd results are caused by the lower resolution simulations used for this h-refinement study suffering
from “non-fatal” instabilities. This simply means an instability of some kind appears to be present without
causing the simulation to blow up. Looking at figure 14, non-fatal instabilities can be found in the following
simulations: P2–802,P3–402, and P4–402. These instabilities cause the linear fit computation to report an
incorrect order of accuracy.
The point of presenting these results is to make the reader aware that care needs to be taken when
performing an h-refinement study using the vortex problem. In most literary articles that use the isentropic
Euler vortex problem for this purpose,9,14,21–23,25–27,29,30 the order of accuracy is computed using the
solution error at some arbitrary point in time on a series of grid resolutions. Here we have shown that this
can be extremely error prone with us incorrectly showing a 3rd order method with 8th order accuracy. Our
incorrect results showed the method in a positive light, but this issue could very easily present a negative
result with an order of accuracy that is less than expected. A common source of negative results is when
a simulation is so overresolved that it encounters machine precision limits. This reasoning is exactly why
the present study was capped at P4 solutions due to P5 and P6 simulations incorrectly reporting poor
performance.
By prescreening the results of all our simulations, we can selectively remove from our h-refinement study
those cases showing any indication there was either too little or too much grid resolution. Since one or more
simulations on the 402and 802grids showed signs of instabilities, we removed all the simulations on those
grids from our h-refinement study leaving only the simulations run on the 1202, 1602, and 2002grids. We
did this to keep the grid levels identical for all polynomial orders. Recomputing the linear fits with this
selectively chosen set of simulations produces the order of accuracy versus time plot shown in figure 15b.
These results are much cleaner than when we blindly used all of our simulations and closer to what is
expected. Initially, the accuracy for each polynomial order is P+1 which is a result commonly found for DG
type methods used with nonlinear equations. As time progresses, the accuracy increases for all polynomial
orders reaching an approximate accuracy of P+ 3 through 100 periods. This clearly verifies that the FR
method achieves some degree of super accuracy, that is accuracy greater than P+ 1, for this problem.
IV.D. Analysis and Recommendations
Based on our experiences with the isentropic Euler vortex problem, the following is a list of general guidelines
and recommendations to keep in mind with its use.
1. If using periodic boundary conditions, make sure that the size of the computational domain is large
enough so that the velocity perturbations are zero at the boundaries. Otherwise, the opposing tangen-
tial velocities across the periodic boundary create an artificial shear layer that does not exist in the
exact analytical solution. The error associated with this shear layer masks the error of the discretization
producing an incorrect total solution error that is larger than expected.
2. When using the vortex problem to perform an h-refinement study, make sure that all of the simulations
used in the study are not under- or overresolved. Both of these situations can cause the computed
order of accuracy to be incorrect.
3. Because the error can change rapidly between multiple simulations, it is also recommended to perform
an h-refinement analysis at several points in time to make sure that the computed order of accuracy
16 of 21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
Figure 13: Time series showing how the density error emanates from the interior of the vortex. All boundaries
use characteristic boundary conditions to prevent errors from accumulating within the computational domain.
17 of 21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
Period
Log10 ( L2 Error )
0 20 40 60 80 100
-10 -10
-8 -8
-6 -6
-4 -4
-2 -2
P2
P3
P4
8021602
12022002
402
Grid SizesPolynomial Order
Figure 14: Density error versus period for all the stationary vortex simulations used to perform an h-
refinement study with the FR method.
is itself accurate. An increased sampling of the accuracy is the only way to ensure that the results are
correct.
4. If propagating the vortex for more than a single period, the numerical error must have a means of
escaping the computational domain in order to get the best results. This is especially critical when
using high-order methods that produce minimal numerical dissipation. When the vortex is propagating,
it takes less than one period for error to recirculate completely and impinge on the vortex from the
opposite direction. We found that setting the boundaries in one direction to some sort of outflow
boundary condition is sufficient to prevent error from accumulating.
5. If any boundaries are periodic, the simulation is not of a single vortex but of an infinite array of coupled
vortices that interact due to the nonlinearity of the Euler equations. The only way to truly decouple
the vortex is to keep the vortex stationary by changing the mean flow velocity to zero and using an
inflow/outflow condition on all domain boundaries. These changes require extra care that sufficient
grid resolution is used due to additional numerical instabilities related to low-speed flows. Provided
this extra care is taken, the stationary, decoupled vortex can provide additional insight into the error
generated by a numerical method.
Upon further examination, the nonlinear instability previously encountered for the isentropic Euler vortex
problem appears to be the confluence of several factors. First, throughout the entirety of this work, we have
been using the correction function that recovers the nodal DG method which is known for producing very little
numerical dissipation. Second, there is zero physical dissipation in the governing Euler equations. Finally,
there is no way for energy/information to truly exit the domain if all the boundaries of the computational
domain are periodic. Combining all of these factors results in a system with little to no ability to dissipate
the error or a means of letting it escape the domain. The error that is produced throughout the simulation,
whether it is from the truncation error of the numerical method or from something as small as machine
roundoff error, will continuously build upon itself until it eventually becomes large enough to affect the
system in some capacity.
If we were instead using a 2nd-order method, say a finite-volume MUSCL (Monotonic Upstream-Centered
Scheme for Conservation Laws) scheme, this problem would not exist because the numerical method provides
sufficient dissipation to suppress the error from accumulating. However, without a very high grid resolution
18 of 21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
Period
Order of Accuracy
0 20 40 60 80 100
3 3
4 4
5 5
6 6
7 7
8 8
P2
P3
P4
(a) Including simulations with signs of instabilities.
Period
Order of Accuracy
0 20 40 60 80 100
3 3
4 4
5 5
6 6
7 7
8 8
P2
P3
P4
(b) Selectively using only stable simulations.
Figure 15: The development of the order of accuracy over time for the stationary vortex using all character-
istic boundaries.
it is unlikely that a 2nd-order finite-volume method will be able to propagate the vortex for very long before
the vortex is dissipated completely, leaving only the underlying mean flow to remain.20
As this problem has become increasingly popular for verifying high-order methods, various literary works
have reported instabilities very similar to what has been shown in the current work.23,27 When encountered
within the family of FR methods, these instabilities have generally been explained as being caused by
polynomial aliasing.23,27 Whereas numerical methods utilizing polynomial basis functions have been known
to encounter aliasing instabilities,13,14,23,27,37–39 the results in the current work, along with those in Ref. 40,
indicate that these instabilities with the vortex problem are more likely related to the problem being ill-posed.
This seems to be a case of mistaken identity simply because aliasing errors are not completely understood
and present an open problem for these methods.
Finally, we should reevaluate our results within the context of the true objective for this test case. The
primary purpose of this canonical problem is to demonstrate the ability of a numerical method to sustain
vortical structures inherent to turbulent flows without contributing unwanted numerical dissipation. Because
the final goal is simulating turbulent flows, the issue of insufficient dissipation solving the Euler equations
is mostly irrelevant due to the physical dissipation within the Navier-Stokes equations. Additionally, the
computational domain for nearly all real-world problems contains some form of boundary condition allowing
for outflow from the domain. The previous results have clearly demonstrated the primary purpose of this
problem with our ability to indefinitely propagate the vortex provided that there is a way for error to leave
the system.
V. Conclusions
The isentropic Euler vortex problem is a simple, canonical test case commonly used to test the accuracy of
an inviscid CFD code. Throughout the many variations of this problem that exist in the literature, several
commonly made assumptions in the problem definition have been found to cause numerical instabilities,
especially when using high-order methods. For the unknowing researcher looking to add this problem to
their CFD toolbox, a list of general guidelines and recommendations has been provided to help prevent these
problems.
After following these recommendations ourselves, we used the isentropic Euler vortex problem to test
the accuracy of our high-order code that is based on the FR method. We were able to successfully verify
its accuracy and found P+ 1 order accuracy through 1 period and super-accuracy, i.e., greater than P+ 1,
through 100 periods. This work shows that vortical flow structures can be accurately predicted by the FR
method with minimal unwanted numerical dissipation introduced into the system.
19 of 21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
References
1DeBonis, J., “Progress Towards Large-Eddy Simulations for Prediction of Realistic Nozzle Systems,” AIAA Journal of
Propulsion and Power, Vol. 23, No. 5, 2007, pp. 971–980.
2Forsythe, J., Squires, K., Wurtzler, K., and Spalart, P., “Detached-Eddy Simulation of Fighter Aircraft at High-Alpha,”
Journal of Aircraft, Vol. 41, No. 2, 2004, pp. 193–200.
3Mavripilis, D., Pelaez, J., and Kandil, O., “Large Eddy and Detached Eddy Simulations Using an Unstructured Multigrid
Solver,” DNS/LES - Progress and Challenges. Proceedings of the Third AFOSR International Conference on DNS/LES ,
Columbus, OH, 2001.
4Moreau, S., Christophe, J., and Roger, M., “LES of the Trailing-Edge Flow and Noise of a NACA0012 Airfoil Near Stall,”
Proceedings of the Summer Program 2008, Center for Turbulence Research, Stanford University/NASA Ames, 2008.
5Kang, S., Iaccarino, G., Ham, F., and Moin, P., “Prediction of Wall-Pressure Fluctuation in Turbulent Flows with an
Immersed Boundary Method,” Journal of Computational Physics, Vol. 228, No. 18, 2009, pp. 6753–6772.
6Huynh, H., “A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods,” AIAA
Paper 2007-4079, Jun 2007.
7Huynh, H., “A Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin for Diffusion,” AIAA
Paper 2009-403, Jan 2009.
8Huynh, H., “High-Order Methods Including Discontinuous Galerkin by Reconstructions on Triangular Meshes,” AIAA
Paper 2011-44, Jan 2011.
9Vermeire, B. C., Cagnone, J.-S., and Nadarajah, S., “ILES Using the Correction Procedure via Reconstruction Scheme,”
AIAA Paper 2013-1001, Jan 2013.
10Vermeire, B. C., Nadarajah, S., and Tucker, P. G., “Canonical Test Cases for High-Order Unstructured Implicit Large
Eddy Simulation,” AIAA Paper 2014-0935, Jan 2014.
11Skarolek, V. and Miyaji, K., “Transitional Flow over a SD7003 Wing Using Flux Reconstruction Scheme,” AIAA Paper
2014-0250, Jan 2014.
12Haga, T., Tsutsumi, S., Kawai, S., and Takaki, R., “Large-Eddy Simulation of a Supersonic Jet Using High-Order Flux
Reconstruction Scheme,” AIAA Paper 2015-0831, Jan 2015.
13Karniadakis, G. and Sherwin, S., Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University
Press, 2nd ed., 2013.
14Hesthaven, J. S. and Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications,
Vol. 54 of Texts in Applied Mathematics, Springer New York, 2008.
15Kopriva, D. A., “Metric Identities and the Discontinuous Spectral Element Method on Curvilinear Meshes,” Journal of
Scientific Computing, Vol. 26, No. 3, 2006, pp. 301–327.
16Kopriva, D. A., Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engi-
neers, Springer Science & Business Media, 2009.
17Roe, P. L., “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes,” Journal of Computational
Physics, Vol. 43, No. 2, 1981, pp. 357–372.
18Huynh, H. T., “Accurate Upwind Methods for the Euler Equations,” SIAM Journal on Numerical Analysis, Vol. 32,
No. 5, 1995, pp. 1565–1619.
19Gottlieb, S. and Shu, C.-W., “Total Variation Diminishing Runge-Kutta Schemes,” Mathematics of Computation, Vol. 67,
No. 221, 1998, pp. 73–85.
20Shu, C.-W., “Essentially Non-oscillatory and Weighted Essentially Non-oscillatory Schemes for Hyperbolic Conservation
Laws,” Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, edited by A. Quarteroni, Vol. 1697 of Lecture
Notes in Mathematics, Springer Berlin Heidelberg, 1998, pp. 325–432.
21Wang, Z. J., Liu, Y., May, G., and Jameson, A., “Spectral Difference Method for Unstructured Grids II: Extension to
the Euler Equations,” Journal of Scientific Computing, Vol. 32, No. 1, 2007, pp. 45–71.
22Wang, Z. and Gao, H., “A Unifying Lifting Collocation Penalty Formulation Including the Discontinuous Galerkin,
Spectral Volume/Difference Methods for Conservation Laws on Mixed Grids,” Journal of Computational Physics, Vol. 228,
No. 21, 2009, pp. 8161–8186.
23Castonguay, P., Vincent, P., and Jameson, A., “Application of High-Order Energy Stable Flux Reconstruction Schemes
to the Euler Equations,” AIAA Paper 2013-686, Jan 2011.
24Vincent, P., Castonguay, P., and Jameson, A., “Insights from von Neumann Analysis of High-Order Flux Reconstruction
Schemes,” Journal of Computational Physics, Vol. 230, No. 22, 2011, pp. 8134–8154.
25Wang, Z., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K.,
Huynh, H., Kroll, N., May, G., Persson, P.-O., van Leer, B., and Visbal, M., “High-order CFD Methods: Current Status and
Perspective,” International Journal for Numerical Methods in Fluids, Vol. 72, No. 8, 2013, pp. 811–845.
26Gao, H. and Wang, Z., “A Conservative Correction Procedure via Reconstruction Formulation with the Chain-Rule
Divergence Evaluation,” J. Comput. Physics, Vol. 232, No. 1, 2013, pp. 7–13.
27Williams, D. M. and Jameson, A., “Nodal Points and the Nonlinear Stability of High-Order Methods for Unsteady Flow
Problems on Tetrahedral Meshes,” AIAA Paper 2013-2830, Jun 2013.
28Witherden, F. D., Farrington, A. M., and Vincent, P. E., “PyFR: An Open Source Framework for Solving Advection-
Diffusion Type Problems on Streaming Architectures Using the Flux Reconstruction Approach,” Computer Physics Commu-
nications, Vol. 185, No. 11, 2014, pp. 3028–3040.
29De Grazia, D., Mengaldo, G., Moxey, D., Vincent, P. E., and Sherwin, S. J., “Connections Between the Discontinuous
Galerkin Method and High-Order Flux Reconstruction Schemes,” International Journal for Numerical Methods in Fluids,
Vol. 75, No. 12, 2014, pp. 860–877.
20 of 21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
30Yu, M., Wang, Z., and Liu, Y., “On the Accuracy and Efficiency of Discontinuous Galerkin, Spectral Difference and
Correction Procedure via Reconstruction Methods,” Journal of Computational Physics , Vol. 259, 2014, pp. 70–95.
31Taylor, M. A., Wingate, B. A., and Bos, L. P., “Several New Quadrature Formulas for Polynomial Integration in the
Triangle,” arXiv:math/0501496v2, 2007.
32Spiteri, R. J. and Ruuth, S. J., “A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization
Methods,” SIAM Journal on Numerical Analysis, Vol. 40, No. 2, 2002, pp. 469–491.
33Carpenter, M. H. and Kennedy, C. A., “Fourth-Order 2N-Storage Runge-Kutta Schemes,” NASA TM-1994-109112, 1994.
34Toro, E. F., Spruce, M., and Speares, W., “Restoration of the Contact Surface in the HLL-Riemann Solver,” Shock waves,
Vol. 4, No. 1, 1994, pp. 25–34.
35Liou, M.-S., “A Sequel to AUSM: AUSM+,” Journal of Computational Physics, Vol. 129, No. 2, 1996, pp. 364–382.
36Edwards, J. R., “A Low-Diffusion Flux-Splitting Scheme for Navier-Stokes Calculations,” Computers & Fluids, Vol. 26,
No. 6, 1997, pp. 635–659.
37Gassner, G. J. and Beck, A. D., “On the Accuracy of High-Order Discretizations for Underresolved Turbulence Simula-
tions,” Theoretical and Computational Fluid Dynamics, Vol. 27, No. 3-4, 2013, pp. 221–237.
38Kirby, R. M. and Karniadakis, G. E., “De-Aliasing on Non-Uniform Grids: Algorithms and Applications,” Journal of
Computational Physics, Vol. 191, No. 1, 2003, pp. 249–264.
39Kirby, R. M. and Sherwin, S. J., “Aliasing Errors Due to Quadratic Nonlinearities on Triangular Spectral/hp Element
Discretisations,” Journal of Engineering Mathematics, Vol. 56, No. 3, 2006, pp. 273–288.
40Spiegel, S., Huynh, H. T., and DeBonis, J. R., “De-Aliasing through Over-Integration Applied to the Flux Reconstruction
and Discontinuous Galerkin Methods,” AIAA Paper 2015-2744, Jun 2015.
41DeBonis, J. R., “A High-Resolution Capability for Large-Eddy Simulation of Jet Flows,” AIAA Paper 2010-5023, Jun
2010.
21 of 21
American Institute of Aeronautics and Astronautics
Downloaded by NASA GLENN RESEARCH CENTER on January 17, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2015-2444
