Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations
ABSTRACT The novel implicit and unconditionally stable, high resolution Total Variation Diminishing (TVD) scheme whose application to steady state calculations is presently examined is a member of a one-parameter family of implicit, second-order accurate systems developed by Harten (1983) for the computation of weak solutions for one-dimensional hyperbolic conservation laws. The scheme will not generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments for a quasi-one-dimensional nozzle problem show that the experimentally determined stability limit correlates exactly with the theoretical stability limit for the nonlinear scalar hyberbolic conservation laws.
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ABSTRACT: SUMMARYA semi-implicit finite element scheme and a Newton-like solver are developed for the stationary compressible Euler equations. Since the Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable, the troublesome antidiffusive part is constrained within the framework of algebraic flux correction. A generalization of total variation diminishing (TVD) schemes is employed to blend the original Galerkin scheme with its nonoscillatory low-order counterpart. Unlike standard TVD limiters, the proposed limiting strategy is fully multidimensional and readily applicable to unstructured meshes. However, the nonlinearity and nondifferentiability of the limiter function makes efficient computation of stationary solutions a highly challenging task, especially in situations when the Mach number is large in some subdomains and small in other subdomains. In this paper, a semi-implicit scheme is derived via a time-lagged linearization of the Jacobian operator, and a Newton-like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. A boundary Riemann solver is used to avoid unphysical boundary states. It is shown that the proposed approach offers unconditional stability, as well as higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense. The overall spatial accuracy of the constrained scheme and the benefits of the new boundary treatment are illustrated by grid convergence studies for 2D benchmark problems. Copyright © 2011 John Wiley & Sons, Ltd.International Journal for Numerical Methods in Fluids 05/2012; 69(1):1 - 28. · 1.35 Impact Factor
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ABSTRACT: Astrophysical fluid dynamical problems rely on efficient numerical solution techniques for hyperbolic and parabolic terms. Efficient techniques are available for treating the hyperbolic terms. Parabolic terms, when present, can dominate the time for evaluating the solution, especially when large meshes are used. This stems from the fact that the explicit time-step for parabolic terms is proportional to the square of the mesh size and can become unusually small when the mesh is large. Multigrid-Newton-Krylov methods can help, but usually require a large number of iterations to converge. Super TimeStepping schemes are an interesting alternative, because they permit one to take very large overall time-steps for the parabolic terms while using only a modest number of explicit time-steps. Super TimeStepping schemes of the type used in astrophysics have, so far, been only first-order accurate in time and prone to instabilities. In this paper, we present a Runge-Kutta method that is based on the recursion sequence for Legendre polynomials, called the RKL2 method. RKL2 is a time-explicit method that permits us to treat non-linear parabolic terms robustly and with large, second-order accurate time-steps. An s-stage RKL2 scheme permits us to take a time-step that is ˜s2 times larger than a single explicit, forward Euler time-step for the parabolic operator. This permits an s-fold gain in computational efficiency over explicit time-step sub-cycling. For modest values of 's', the advantage can be substantial. The stability properties of the new schemes are explored and they are shown to be stable and positivity preserving for linear operators. We document the method as it is applied to the anisotropic thermal conduction operator for dilute, magnetized, astrophysical plasmas. Implementation-related details are discussed. The RKL2 Super TimeStepping scheme has been implemented in the RIEMANN code for computational astrophysics. We explain the method for picking an s-stage RKL2 scheme for the parabolic terms and show how it can be integrated with a hyperbolic system solver. The method's simplicity makes it very easy to retrofit the s-stage RKL2 scheme to any problem with a parabolic part when a well-formed spatial discretization is available. Several stringent test problems involving thermal conduction in astrophysical plasmas are presented and the method is shown to perform robustly and efficiently on all of them.Monthly Notices of the Royal Astronomical Society 05/2012; 422(3):2102-2115. · 5.52 Impact Factor
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ABSTRACT: We study buoyant displacement flows in a plane channel with two fluids in the long-wavelength limit in a stratified configuration. Weak inertial effects are accounted for by developing a weighted residual method. This gives a first-order approximation to the interface height and flux functions in each layer. As the fluids are shear-thinning and have a yield stress, to retain a formulation that can be resolved analytically requires the development of a system of special functions for the weight functions and various integrals related to the base flow. For displacement flows, the addition of inertia can either slightly increase or decrease the speed of the leading displacement front, which governs the displacement efficiency. A more subtle effect is that a wider range of interface heights are stretched between advancing fronts than without inertia. We study stability of these systems via both a linear temporal analysis and a numerical spatiotemporal method. To start with, the Orr–Sommerfeld equations are first derived for two generalized non-Newtonian fluids satisfying the Herschel–Bulkley model, and analytical expressions for growth rate and wave speed are obtained for the long-wavelength limit. The predictions of linear analysis based on the weighted residual method shows excellent agreement with the Orr–Sommerfeld approach. For displacement flows in unstable parameter ranges we do observe growth of interfacial waves that saturate nonlinearly and disperse. The observed waves have similar characteristics to those observed experimentally in pipe flow displacements. Although the focus in this study is on displacement flows, the formulation laid out can be easily used for similar two-layer flows, e.g. co-extrusion flows.Journal of Fluid Mechanics 09/2013; 731. · 2.18 Impact Factor
JOURNAL OF COMPUTATIONAL PHYSICS 57, 327-360 (1985)
Implicit Total Variation Diminishing
H. C. YEE AND R. F. WARMING
NASA Ames Research Center, Moffett Field, California
Tel Aviv University, Tel Aviv, and New York University, New York
Received August 25, 1983
The application of a new implicit unconditionally stable high-resolution TVD scheme to
calculations is examined. It is a member of a one-parameter family of explicit and
implicit second-order accurate schemes developed by Harten for the computation of weak
solutions of one-dimensional hyperbolic conservation laws. This scheme is guaranteed not to
generate spurious oscillations for a nonlinear scalar equation and a constant coethcient
system. Numerical experiments show that this scheme not only has a fairly rapid convergence
rate, but also generates a highly resolved approximation to the steady-state solution. A
detailed implementation of the implicit scheme for the one- and two-dimensional compressible
inviscid equations of gas dynamics is presented. Some numerical computations of one- and
two-dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this
0 1985 Academic Press, Inc.
Conventional shock capturing schemes for the solution of nonlinear hyperbolic
conservation laws are linear and &-stable (stable in the &-norm) when considered
in the constant coefficient case [l]. There are three major difficulties in using such
schemes to compute discontinuous solutions of a nonlinear system, such as the
compressible Euler equations:
(i) Schemes that are second (or higher) order accurate may produce
oscillations wherever the solution is not smooth.
(ii) Nonlinear instabilities may develop in spite of the &-stability in the con-
stant coefficient case.
(iii) The scheme may select a nonphysical solution.
It is well known that monotone conservative difference schemes always converge
and that their limit is the physical weak solution satisfying an entroy inequality.
1985 by Academic
in any form reserved.
Copyright 0 Press, Inc.
All rights of reproduction
YEE, WARMING, AND HARTEN
Thus monotone schemes are guaranteed not to have difficulties (ii) and (iii).
However, monotone schemes are only first-order accurate. Consequently, they
produce rather crude approximations whenever the solution varies strongly in space
When using a second- (or higher) order accurate scheme, some of these dif-
ficulties can be overcome by adding a hefty amount of numerical dissipation to the
scheme. Unfortunately, this process brings about an irretrievable loss of infor-
mation that exhibits itself in degraded accuracy and smeared discontinuities. Thus,
a typical complaint about conventional schemes which are developed under the
guidelines of linear theory is that they are not robust and/or not accurate enough.
To overcome the difficulties, we consider a new class of schemes that is more
appropriate for the computation of weak solutions (i.e., solutions with shocks and
contact discontinuities) of nonlinear hyperbolic conservation laws. These schemes
are required (a) to be total variation diminishing in the nonlinear scalar case and
the constant coefficient system case [2, 31 and (b) to be consistent with the conser-
vation law and an entropy inequality [4, 61. The first property guarantees that the
scheme does not generate spurious oscillations. We refer to schemes with this
property as total variation diminishing (TVD) schemes (or total variation non-
increasing, TVNI, ). The latter property guarantees that the weak solutions are
physical ones. Schemes in this class are guaranteed to avoid difficulties (ik(iii)
The class of TVD schemes contains monotone schemes, but is significantly larger
as it includes second-order accurate schemes. Existence of second-order accurate
TVD schemes was demonstrated in [2, 3, 7, 81. Unlike monotone schemes, TVD
schemes are not automatically consistent with the entropy inequality. Consequently,
some mechanism may have to be explicitly added to a TVD scheme to enforce the
selection of the physical solution. In [2, 93, Harten and Harten and Hyman
demonstrate a way of modifying a TVD scheme to be consistent with an entropy
In [ 10, 111, we have examined the application of an explicit second-order
accurate TVD scheme  to steady-state calculations. Numerical experiments
show that this explicit scheme generates nonoscillatory, highly accurate steady-state
To retain the characteristic of highly resolved steady-state solutions by explicit
second-order accurate TVD schemes without the disadvantage of slow convergence
rate of explicit schemes, we considered in [lo]
(1) First, obtain an approximation to the steady state by using a conventional
implicit scheme, and then use a second-order accurate TVD scheme as a “post-
processor.” (2) Use a first-order accurate implicit scheme in delta-formulation and
replace the explicit operator by an explicit second-order accurate TVD scheme.
We have found (in one dimension) that both these strategies reduce the overall
computational effort needed to obtain the steady-state solution of the explicit
second-order accurate TVD scheme. Alternative (1) is a possible way of speeding
up the convergence process by providing a better initial condition for the explicit
the following two possibilities:
second-order accurate TVD scheme. Alternative (2) can be viewed as a relaxation
procedure to the steady-state solution. Numerical experiments of [lo] show that
the computational effort is not drastically decreased, although the stability limit is
higher than the explicit counterpart.
Recently, Harten  has extended the class of explicit TVD schemes to a more
general category which includes a one-parameter family of implicit second-order
accurate schemes. Included in this class are the commonly used time-differencing
schemes such as the backward Euler and the trapezoidal formula.
This paper is a sequel to [lo]. Here, we investigate the application to steady-
state calculations of this newly developed implicit second-order accurate scheme
that is unconditionally TVD. This scheme is guaranteed not to generate spurious
oscillations for one-dimensional nonlinear scalar equations and constant coefficient
systems. Numerical experiments show that this scheme has a fairly rapid con-
vergence rate, in addition to generating a highly resolved approximation to the
steady-state solution. We remark that all of the analysis on the new scheme is for
the initial value problem. The numerical boundary conditions are not included.
In the present paper, we stress applications rather than theory, and we refer the
interested reader to [2, 31 for more theoretical details. In the next section, we will
briefly review the notion of TVD schemes and describe the construction of the
second-order accurate TVD scheme from a first-order accurate one for scalar one-
dimensional hyperbolic conservation laws. The generalization to one-dimensional
hyperbolic systems will be described in Section 3. A description of the algorithm
and numerical results for the one- and two-dimensional compressible inviscid
equations of gas dynamics will be presented in Sections 4 and 5.
2. TVD SCHEMES FOR ONE-DIMENSIONAL SCALAR
HYPERBOLIC CONSERVATION LAWS
Several techniques for the construction of nonlinear, explicit, second-order
accurate, high-resolution, entropy satisfying schemes for hyperbolic conservation
laws have been developed in recent years. See, for example, van Leer , Colella
and Woodhard , and Harten . From the standpoint of numerical analysis,
these schemes are TVD for nonlinear scalar hyperbolic conservation laws and for
constant coefficient hyperbolic systems. TVD schemes are usually rather com-
plicated to use compared to the conventional shock-capturing methods such as
variants of the Lax-Wendroff scheme.
In , Harten introduced the notion of implicit TVD schemes. To keep this
paper somewhat self-contained, we will review the construction of the backward
Euler TVD schemes for the initial value problem. This is the only unconditionally
stable TVD scheme belonging to the one-parameter family of TVD schemes con-
sidered in . Before we proceed with the description of the construction, we will
first give preliminaries on the definition of explicit and implicit TVD schemes and
show a few examples.
YEE, WARMING, AND HARTEN
2.1. Explicit TVD Schemes
Consider the scalar hyperbolic conservation law
at+- ax ’
where a(u) = aflau is the characteristic speed. A general three-point explicit dif-
ference scheme in conservation form can be written
u? + l = ui” - ncjy+ I,* - 3;- l/2)>
where 37+ 1,2 = f( ~7, uj’+ r ), I= At/Ax, with At the time step, and Ax the mesh size.
Here, ~7 is a numerical solution of (2.1) at x = j Ax and t = n At and j’ is a
numerical flux function. We require the numerical flux function 3 to be consistent
with the conservation law in the following sense:
3c5 Uj) =“I@,).
Consider a numerical scheme with numerical flux functions of the form
a ,+ l/2 =
t-G+ I -f,)lA.j+ 1/2u,
A,+ I/~u # 0,
Here Q is a function of uj+ 1,2 and 1. The function Q is sometimes referred to as the
coefficient of numerical viscosity. Figure 2.1 shows some examples for the possible
choice of Q. Three familiar schemes with the numerical fluxes of the form (2.4) are
-. 5 -6
s .5 1.0
FIG. 2.1. Sample of the Q(z) functions.
IMPLICIT TVD SCHEMES
(a) A form of the Lax-Wendroff (L-W) scheme with
where Q(Qj+ 112) = l(aj+ 1,d2.
(b) Lax-Friedrichs (L-F) scheme with
where Q(aj+ 1,2) = l/k
(c) A generalization of the Courant-Isaacson-Rees (GCIR) scheme with
CJ’j+f,+l- lUj+1/21 Aj+1/2ul,
where Q(Uj+ 1,~) = lUj+ 4.
We define the total variation of a mesh function u to be
TV(u)= f luj+l -ujI = f
We say that the numerical scheme (2.2) is TVD if
It can be shown that a sufficient condition for (2.2) together with (2.4) to be a
TVD scheme is ,
AC,< 1,2 = '1 2 Eeuj+
l/2 + Q(q+ 1,211 L 0,
‘CJ”+ l/2 =i [uj+ l/2 + Q(uj+ 1,211 > 0,
‘Cc; l/2 + CA l/2) = nQ(aj+ 112) G 1.
Applying condition (2.11) and/or (2.10) to the above three examples, it can be
easily shown that the L-W scheme is not a TVD scheme, and the latter two
schemes are TVD schemes. Note that there is a further distinction between the L-F
scheme and GCIR scheme: the L-F scheme is consistent with an entropy inequality
whereas the GCIR is not .
It should be emphasized that condition (2.11) is only a sufficient condition; i.e.,
schemes that fail this test might be still TVD. The L-W scheme, besides failing con-
dition (2.1 l), does not satisfy (2.10).
YEE, WARMING, AND HARTEN
2.2. Implicit TVD Schemes
Now we consider a one-parameter family of three-point conservative schemes of
24” + l + lJj(fy;1’,2
-f;‘lg = q - 41 - r1)(fy+ I,2 -fp l/2)?
where q is a parameter, A= At/Ax, f;+ 1,2 = f( u,“, u,“+ r ), f;:r& = f( U; + I, u,“:,’ ), and
f(u,, uj+,) is th e numerical flux (2.4). This one-parameter family of schemes con-
tains implicit as well as explicit schemes. When q = 0, (2.12) reduces to (2.2), the
explicit method. When q # 0, (2.12) is an implicit scheme. For example: if u = t, the
time differencing is the trapezoidal formula, and if q = 1, the time differencing is the
backward Euler method. To simplify the notation, we will rewrite (2.12) as
where L and R are the finite-difference operators
A suflicient condition for (2.12) to be a TVD scheme is that
TV( R . u) 6 TV(u),
TV( L . v) B TV(o).
A sufficient condition for (2.15) is the CFL-like restriction
where aj + 1,2 is defined in (2.5). For a detailed proof of (2.15) and (2.16), see .
Observe that the backward Euler implicit scheme, q = 1 in (2.12), is unconditionally
TVD, while the trapezoidal formula, q = 4, is TVD under the CFL-like restriction of
2. The forward Euler explicit scheme, r~ = 0 or (2.2), is TVD under the CFL restric-
tion of 1. We remark that three-point conservative TVD schemes of the form (2.12)
are generally first-order accurate in space. When q = 1, the scheme is second-order
accurate in time.
2.3. First-Order Accurate Backward Euler Implicit TVD Scheme
In this paper, we are only interested in efficient high-resolution time-dependent
methods for steady-state calculations. The backward Euler implicit TVD scheme is
the best choice in this one-parameter family of TVD schemes. Therefore, we will
only review the proof that the backward Euler scheme is unconditionally TVD. In
Section 2.4, we will describe the technique of converting the first-order accurate
unconditionally TVD scheme (2.12) with q = 1 into a second-order accurate one.
IMF’LICIT TVD SCHEMES
The backward Euler three-point scheme in conservative form can be written as
ui”’ l+ n(f;;;,, -f;$,, = 24;.
For the purpose of this paper, the function Q(z) in (2.4) is chosen to be
which is a nonvanishing, continuously differentiable approximation to 1~1. This is
one way of modifying Q in (2.8) so that scheme (2.2) together with (2.8) is an
entropy satisfying TVD scheme . For convenience, we introduce the notation
C’(z) = $[Q(z) f z]
and note that
for all z.
Using (2.5) and (2.18a), we can rewrite the numerical fluxes f;.+ 1,2 in (2.4) as
j;.- 112 =fi- C+ (aj- 112) A,- 112~.
It follows from (2.19) that (2.17) can be written in the form
u?+’ - J.C-(a~~~,,) A,, I,2~n+1 + K’(uJ’?~,,)
Aj- 1,2~“+1 = u;.
Now, if 6 = 0 in (2.17b), then C’(z) = (lzl f z)/2, and (2.20) is a first-order
accurate, upstream differencing, backward Euler implicit scheme. Equation (2.17)
differs from the upstream spatial differencing (with 6 =0) by the addition of a
numerical viscosity term with a coefftcient 6 > 0.
We show now that C’(z)30 implies that the scheme (2.17) is unconditionally
TVD (i.e., condition (2.10) is satisfied, independent of the value of A = At/Ax in
To see that, we subtract (2.20) at j from (2.20) at j + 1 and get after rearranging
=Aj+ l/2 lln + lCj”- l/ZAj-
l/Z” n+l +K,~3/2Aj+3,2~n+?
Here Cj$ 1,2 = C * (a;:$, ). Next we take the absolute value of (2.21a) and use
(2.18b) and the triangle inequality to obtain
G lAj+ I/ZU~I + ‘CjT 112 IAj- 112~
n+ll +~C~3/21Aj+3/2Un+‘I. (2.21b)
YEE, WARMING, AND HARTEN
Rearranging terms, we get
/A I+ 112 u”+‘I d Pi+,,, ‘“l+‘CC,~~/21’j+~/2~n+‘I-C,~~/~IAj+~,~~nf’l]
IA,+ 1/2u ‘+‘I 6 IAj+l/2U”l +%(Ej+l-Zj),
Summing (2.21d) fromj=
backward Euler implicit scheme is unconditionally TVD.
-co toj= +a~, we obtain (2.10); thus proving that our
2.4. Conversion to Second-Order Accurate Scheme
Next, we want to briefly review the design principle behind the construction of
second-order accurate TVD schemes. This is a rather general technique to convert a
three-point first-order accurate (in space) TVD scheme (2.12) into a five-point
second-order accurate (in both time and space, or just space) TVD scheme of the
same generic form. The design of high-resolution TVD schemes rests on the fact
that the exact solution to (2.1) is TVD due to the phenomenon of propagation
along characteristics, and is independent of the particular form of the flux f(u) in
(2.1). Similarly, the first-order accurate scheme is TVD subject only to the CFL-like
restriction (2.16) independent of the particular form of the flux. Thus to achieve
second-order accuracy while retaining the TVD property, we use the original TVD
scheme with an appropriately modified flux (f + g), i.e.,
Y,+ l/2 =
k,+ I - g,)lAj+ (/2u,
Aj+ 1/2u + 0,
The requirements on g are: (1) The function g should have a bounded y in
(2.22~) so that (2.22a) is TVD with respect to the modified flux (f+ g). (2) The
modified scheme should be second-order accurate (except at points of extrema). In
[2, 33, Harten devised a recipe for g that satisfies the above two requirements. We
will use this particular form of g for the discussion here. It can be written
IMPLICIT TVD SCHEMES
with oj+ 1,2 = o(aj+ i,J and we choose
c(z) = Q(z) > 0
for steady-state applications. It has the property that the steady-state solution is
independent of At. Or, we choose
CT(z) = i@(z) + AZ*) > 0 (2.22f)
for time-accurate calculations. Note that if a(z) = (Q(z) + Lz*)/2, then (2.22) is
second-order accurate in both time and space . For transient calculations,
second-order accurate in time is preferred.
The form of g in (2.22d) satisfies the relations 
gj=gt"j-l, uj3 uj+lh
g(u, 6 u) = 0,
IYj+ l/21 = lgj+ l- gjlll”j+ 1 -#jl G d”j+ I/2),
Relation (2.23a) shows that the modified numerical flux (2.22b) is consistent with
f(u). Relation (2.23b) shows that the mean-value characteristic speed rj+ ,,z (2.22~)
induced by the flux g is uniformly bounded. Relation (2.23~) implies that (2.22b) is
second-order accurate in space. The form of g appears more complicated than it
really is. The various test functions in (2.22d) can be viewed as an automatic way of
controlling the numerical flux function so that (2.22) is TVD.
The scheme (2.22) can be rewritten in the form (2.20) as
where C’(a + y)J’:;,* - C’(aj”:&
instead of a. The modified scheme (2.22) is of the same generic form as the original
first-order scheme (2.17). Therefore (2.22) is an upstream differencing scheme with
respect to the characteristic field (a + y). Moreover, we have the relation
i.e., C’ is now a function of (a + y)
sign(a + y) = sign(u)
for IzI 2 6, with z = a or (a+ y) in (2.17b). Hence (2.24) is also an upstream dif-
ferencing scheme with respect to the original characteristic field a(u).
Because of (2.23a), the numerical flux (2.22b) of the second-order accurate TVD
scheme depends on four points, i.e., yj+ 1,2 = T(Tcu~-
is formally a live-point scheme. We note, however, that
, , uj, uj+ 1, uj+ *), and thus (2.22)
lb, 4 4 WI = f(u)
for all v and w. Hence, for practical purposes, such as numerical boundary con-
ditions, (2.22) can be regarded as essentially a three-point scheme.
YEE, WARMING, AND HARTEN
We turn now to examine the behavior of TVD schemes around points of
extrema, by considering their application to data, where
u/- 1 <"j=u,+I
In this case gj = g, + L = 0 in (2.22d), and thus the numerical flux (2.22b) becomes
identical to that of the original first-order accurate scheme (2.4); consequently, the
truncation error of (2.22) deteriorates to O((dx)‘) at j and j+ 1. This behavior is
common to all TVD schemes. Thus, for a second-order accurate scheme to be
TVD, it has to have a mechanism that switches itself into a first-order accurate
TVD scheme at points of extrema. Because of the above property, second-order
accurate TVD schemes are genuinely nonlinear (i.e., they are nonlinear even in the
constant coefficient case).
Extension of the one-parameter family of three-point TVD schemes (2.12) to
second-order TVD schemes follows the same procedure except (2.22f) becomes
a(z) = @(z) + /I(q - ;, z2. (2.28)
2.5. Enhancement of Resolution by Artificial Compression
The technique to convert the first-order accurate TVD scheme (2.12) into a
second-order accurate one is closely related to the concept of artificial compression
Truncation error analysis shows that the first-order accurate scheme (2.12) is a
second-order accurate approximation to solutions of the modified equation
where o(a) is defined in (2.22e) or (2.28). We note that the CFL-like restriction
(2.16) implies that o(a) 2 0; thus, the right-hand side of (2.29) is a viscosity term.
Hence the first-order accurate TVD scheme (2.12) is a better approximation to the
viscous equation (2.29) than it is to the original conservation law.
We obtain a second-order approximation to au/at + af/ax = 0 by applying the
first-order scheme (2.12) to the modified flux (f+ g), where g is an approximation
to the right-hand side of (2.29); i.e.,
g = dxa(a) g + O((fl~)~). (2.30)
The application of the first-order scheme to (f + g) has the effect of canceling the
error due to the numerical viscosity to O((~X)~); thus g is an “anti-diffusion” flux.
If we apply the first-order TVD scheme to (f+ (1 + 0) g), 0 > 0, rather than to
(f+ g), we find that the resolution of discontinuities improves with increasing 0.
This observation allows us to use the notion of artificial compression to enhance
the resolution of discontinuities computed by the second-order accurate TVD
IMPLICIT TVD SCHEMES
scheme (2.22). This is done by increasing the size of g in (2.22d) by adding a term
that is O((Ax)*) in regions of smoothness, e.g.,
gj=(l +COOj) gj,
Using gj (2.31) instead of g, makes the numerical characteristic speed more con-
vergent, and therefore improves the resolution of computed shocks. Since
13 = O(dx), this change does not adversely affect the order of accuracy of the
scheme. See  for more details. From numerical experiments, o = 2 seems to be a
We remark that applying too much artificial compression in a region of expan-
sion (i.e., divergence of the characteristic field a = afl&) may result in violation of
the entropy condition. Hence when applying artificial compression, we have to
either turn it off in regions of expansion or limit the size of o in (2.31a), say, by the
value that makes (2.22) with (2.31a) third-order
accurate (in regions of
2.6. Linearized Version of the implicit TVD Scheme
To solve for zJ+’ for the first- or second-order implicit scheme, we have to solve
a set of nonlinear algebraic equations. To overcome this obstacle, we will present a
way of linearizing the implicit TVD scheme. The method will destroy the conser-
vative property but preserve its unconditionally TVD property. We will refer to this
method as the linearized nonconservative implicit (LNI) form. The LNI form is
mainly useful for steady-state calculations, since the scheme is only conservative
after the solution reaches steady state. On the other hand, we have the advantage of
stability and TVD of an unlimited CFL number. Note that the procedure of obtain-
ing the LNI form is applicable to both the first- and second-order accurate implicit
TVD schemes. We will discuss the LNI for the second-order accurate one. To get
the LNI for the first-order accurate TVD scheme, we simply set g= y =0 in the
The LNI form is obtained simply by replacing the coefficients (C+ )n+ ’ in (2.24)
by (C’ )“, i.e.,
Since C’ > 0, it follows from (2.21) that (2.32) is unconditionally TVD.
In delta form notation, (2.32) can be rewritten as
= -x7;+ l/2 -JI:-
YEE, WARMING, AND HARTEN
where the left-hand side equals
with d-=u?+‘--uU” and A.
I+ li2d=dit I -d,. Rearranging terms, we get
E, = -K+(u+y)i”p1,2,
E, = -AC (a + y);, ,,2.
Here, 3;+ 1/2 is (2.22b))(2.22e) calculated at the time level n. It follows from (2.22b)
and (2.33a) that the steady-state solution of (2.33) is
(i) consistent with the conservation form, and
(ii) a spatially second-order accurate approximation to the steady state of
the partial differential equation
(iii) Independent of the time-step At used in the iterations.
Moreover, the iteration matrix associated with (2.34) is a diagonally dominant,
tridiagonal matrix. Note that this linearized construction is not trivial, since the
second-order method is a live-point scheme. Normally the matrix associated with
(2.34) could have been a pentadiagonal matrix. As mentioned before, (2.32) or
(2.34) is not in conservation form and therefore should not be used to approximate
time-dependent solutions (transient solutions). However, it is a suitable scheme for
the calculation of steady-state solutions.
We can also obtain another TVD linearized form by setting y = 0 in (2.34), i.e.,
E, = -AC + (a;- ,,*),
Js = --AC (a,“, 1,2).
Scheme (2.35) is spatially first-order accurate for the implicit operator and spatially
second-order accurate for the explicit operator. It can be shown that (2.35) is still
3. GENERALIZATION TO ONE-DIMENSIONAL
HYPERBOLIC SYSTEM OF CONSERVATION LAWS
In the present state of development, the concept of TVD schemes, like monotone
schemes, is only defined for nonlinear scalar conservation laws or constant coef-
ficient hyperbolic systems. The main difficulty stems from the fact that, unlike the
scalar case, the total variation in x of the solution to the system of nonlinear con-
servation laws is not necessarily a monotonic decreasing function of time. The total
variation of the solution may actually increase at moments of interaction between
waves. Not knowing a diminishing functional that bounds the total variation in x in
the system case, makes it impossible to fully extend the theory of the scalar case to
the system case. What we can do at the moment is to extend the new scalar TVD
scheme to system cases so that the resulting scheme is TVD for the “locally frozen”
constant coefficient system. To accomplish this, we define at each point a “local”
system of characteristic fields. This extension technique is a somewhat generalized
version of the procedure suggested by Roe .
Now, we briefly describe the above approach of extending the second-order
accurate TVD schemes to hyperbolic systems of conservation laws
Here U and F(U) are column vectors of m components and A(U) is the Jacobian
matrix. The assumption that (3.1) is hyperbolic implies that A(U) has real eigen-
values a’(U) and a complete set of right eigenvectors R’(U), I= l,..., m. Hence the
R(U) = (R’(U),..., R”(U))
is invertible. The rows L’(U),..., L”(U) of R(U) ~ ’ constitute an orthonormal set of
left eigenvectors of ,A( U); thus
R- 'AR = diag(a’).
Here diag(a’) denotes a diagonal matrix with diagonal elements a’.
We define characteristic variables W with respect to the state U by
In the constant coefficient case, (3.1) decouples into m scalar equations for the
a’ = constant.
This offers a natural way of extending a scalar scheme to a constant coefficient
YEE, WARMING, AND HARTEN
system by applying it “scalarly” to each of the m scalar characteristic equations
denote some symmetric average of U, and U,, , (to be discussed
Let a:* ‘12, qc l/2, q!, l/2 denote the respective quantities of a’, R’, L’ related to
A( U,, ,j2). Let w’ be the vector elements of W, and let c(i+ ,,2 = M$+, - wj be the
component of A,, 1,2 U = U,, , - U, in the Ith characteristic direction; i.e.,
With the above notation, we can apply scheme (2.22) scalarly to each of the
locally defined (frozen coe$j$ccient ) characteristic variables of (3. I ) as
uy + ’ + 1*@y;“=,;, - Fy,j,) = u,“, (3.7a)
Fj+ 112 = ’ ~(J’j+f’j+~)+i f Cg~+g~+,-Q~~~+,i2+~~+,,2)~~+,,21R~+,,2,
gf=S.maxCO, min(~~+,,21~~+,,21, S~O;~,,,CX~,,,)],
S = skn(uj+
- s:Yu; + I/2>
a:+ l/2 # 09
a;, ,,2 = 0.
Here c$+ ,,2 = c(af+ ,,2), where a(z) is (2.22e) and a:+ ,,2 is (3.6). The corresponding
ii in (2.3 1) for the added artificial compression term is
I@+ I,21 + I+ I,21
The w’ can be different from one characteristic field to another.
Similarly, we generalize the LNI form (2.33) to the system case by
[I-‘(,< ,/>A,+ l/2 + iJ,T 1/2Aj- l/21( U”+ ’ - Un) = -A[?+ l,2 - q.. ,,2]
IMPLICIT TVD SCHEMES
l/2 = R&
Dj= ,,“‘I - Uj’,
l/2 diag(C'(a'+ Y'):, ,,2)(R ~ 'I,"+ ,,2,
where the left-hand side of (3.8a) is equal to
Dj- ‘Jjy 1/2Aj+ 1120 + ‘Jj? 1/2Aj- I/zD,
In the constant coefficient case where A(U) = constant, both (3.7) and (3.8) are
TVD by construction. However, they are not identical; Eq. (3.7) is fully nonlinear
while (3.8) is a version with a linearized left-hand side.
Note that the total variation for the vector mesh function U of the constant coef-
ficient case is defined as
WV= f f
j= -cc ,=I
A particular form of averaging in (3.5) is essential if we require the scheme (3.7)
for m = 1 to be identical to the scalar scheme of Section 2, since we have to choose
(3.5) so that u,!+ 1,2 is the same as the mean value in Eq. (2.5). This can be accom-
plished by taking the eigenvalues a:+ 1,2 and the eigenvectors Rj+ 1,2 in (3.2) to be
those of A( U,, U,, I ), where A( Uj, U,, , ) is the mean value Jacobian. This matrix
(i) F(U)--F(;O=A(U, V)(U- V),
(ii) A(U, U)=A(U),
(iii) A( U, V) has real eigenvalues and a complete set of eigenvectors.
Roe [ 151 constructs a mean value Jacobian for the Euler equations of gas
dynamics of the form A( U, V) = A( Y( U, I’)), where !P( U, V) is some particular
average. We will discuss Roe’s mean value Jacobian in the next two sections.
OF GAS DYNAMICS
In this section we describe how to apply the implicit TVD scheme (3.8) to the
compressible inviscid equations of gas dynamics (Euler equations). Included in this