Implicit Total Variation Diminishing (TVD) schemes for steadystate 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 oneparameter family of implicit, secondorder accurate systems developed by Harten (1983) for the computation of weak solutions for onedimensional hyperbolic conservation laws. The scheme will not generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments for a quasionedimensional 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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Page 1
JOURNAL OF COMPUTATIONAL PHYSICS 57, 327360 (1985)
Implicit Total Variation Diminishing
for SteadyState
(TVD) Schemes
Calculations
H. C. YEE AND R. F. WARMING
NASA Ames Research Center, Moffett Field, California
AND
A. HARTEN
Tel Aviv University, Tel Aviv, and New York University, New York
Received August 25, 1983
The application of a new implicit unconditionally stable highresolution TVD scheme to
steadystate
calculations is examined. It is a member of a oneparameter family of explicit and
implicit secondorder accurate schemes developed by Harten for the computation of weak
solutions of onedimensional 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 steadystate solution. A
detailed implementation of the implicit scheme for the one and twodimensional compressible
inviscid equations of gas dynamics is presented. Some numerical computations of one and
twodimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this
new scheme.
0 1985 Academic Press, Inc.
1. INTRODUCTION
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.
327
00219991/85 $3.00
1985 by Academic
in any form reserved.
Copyright 0 Press, Inc.
All rights of reproduction
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YEE, WARMING, AND HARTEN
Thus monotone schemes are guaranteed not to have difficulties (ii) and (iii).
However, monotone schemes are only firstorder accurate. Consequently, they
produce rather crude approximations whenever the solution varies strongly in space
or time.
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, [2]). The latter property guarantees that the weak solutions are
physical ones. Schemes in this class are guaranteed to avoid difficulties (ik(iii)
mentioned above.
The class of TVD schemes contains monotone schemes, but is significantly larger
as it includes secondorder accurate schemes. Existence of secondorder 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
inequality.
In [ 10, 111, we have examined the application of an explicit secondorder
accurate TVD scheme [2] to steadystate calculations. Numerical experiments
show that this explicit scheme generates nonoscillatory, highly accurate steadystate
solutions.
To retain the characteristic of highly resolved steadystate solutions by explicit
secondorder 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 secondorder accurate TVD scheme as a “post
processor.” (2) Use a firstorder accurate implicit scheme in deltaformulation and
replace the explicit operator by an explicit secondorder accurate TVD scheme.
We have found (in one dimension) that both these strategies reduce the overall
computational effort needed to obtain the steadystate solution of the explicit
secondorder 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:
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IMPLICIT
TVD SCHJMES
329
secondorder accurate TVD scheme. Alternative (2) can be viewed as a relaxation
procedure to the steadystate 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 [3] has extended the class of explicit TVD schemes to a more
general category which includes a oneparameter family of implicit secondorder
accurate schemes. Included in this class are the commonly used timedifferencing
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 secondorder accurate scheme
that is unconditionally TVD. This scheme is guaranteed not to generate spurious
oscillations for onedimensional 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
steadystate 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
secondorder accurate TVD scheme from a firstorder accurate one for scalar one
dimensional hyperbolic conservation laws. The generalization to onedimensional
hyperbolic systems will be described in Section 3. A description of the algorithm
and numerical results for the one and twodimensional compressible inviscid
equations of gas dynamics will be presented in Sections 4 and 5.
2. TVD SCHEMES FOR ONEDIMENSIONAL SCALAR
HYPERBOLIC CONSERVATION LAWS
Several techniques for the construction of nonlinear, explicit, secondorder
accurate, highresolution, entropy satisfying schemes for hyperbolic conservation
laws have been developed in recent years. See, for example, van Leer [7], Colella
and Woodhard [12], and Harten [2]. 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 shockcapturing methods such as
variants of the LaxWendroff scheme.
In [3], Harten introduced the notion of implicit TVD schemes. To keep this
paper somewhat selfcontained, 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 oneparameter family of TVD schemes con
sidered in [3]. 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.
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YEE, WARMING, AND HARTEN
2.1. Explicit TVD Schemes
Consider the scalar hyperbolic conservation law
au afw=,
at+ ax ’
(2.1)
where a(u) = aflau is the characteristic speed. A general threepoint explicit dif
ference scheme in conservation form can be written
u? + l = ui”  ncjy+ I,*  3; l/2)>
J
(2.2)
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@,).
(2.3)
Consider a numerical scheme with numerical flux functions of the form
&+I,*=+ [~+++1Q(aj+1/2)Aj+1/2~1,
(2.4)
wherefi=f(u,),dj+,,,u=u,+,u,, and
a ,+ l/2 =
tG+ I f,)lA.j+ 1/2u,
A,+ I/~u # 0,
~t~+,,~u=O.
4yj),
(2.5)
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
0
z
s .5 1.0
FIG. 2.1. Sample of the Q(z) functions.
Page 5
IMPLICIT TVD SCHEMES
331
(a) A form of the LaxWendroff (LW) scheme with
~+1/2=~Cfj+fj+1Iz(uj+,/2)2dj+1/2ul,
where Q(Qj+ 112) = l(aj+ 1,d2.
(b) LaxFriedrichs (LF) scheme with
V6)
(2.7)
where Q(aj+ 1,2) = l/k
(c) A generalization of the CourantIsaacsonRees (GCIR) scheme with
fj+*,2=$
CJ’j+f,+l lUj+1/21 Aj+1/2ul,
(2.8)
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
lA,+1/2~l.
(2.9)
j= m
j= cc
We say that the numerical scheme (2.2) is TVD if
TV(u”+‘)6TV(u”).
(2.10)
It can be shown that a sufficient condition for (2.2) together with (2.4) to be a
TVD scheme is [2],
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.
(2.1 la)
(2.11b)
(2.1 lc)
Applying condition (2.11) and/or (2.10) to the above three examples, it can be
easily shown that the LW scheme is not a TVD scheme, and the latter two
schemes are TVD schemes. Note that there is a further distinction between the LF
scheme and GCIR scheme: the LF scheme is consistent with an entropy inequality
whereas the GCIR is not [6].
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 LW scheme, besides failing con
dition (2.1 l), does not satisfy (2.10).
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YEE, WARMING, AND HARTEN
2.2. Implicit TVD Schemes
Now we consider a oneparameter family of threepoint conservative schemes of
the form
24” + l + lJj(fy;1’,2
I
f;‘lg = q  41  r1)(fy+ I,2 fp l/2)?
(2.12)
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 oneparameter 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
L.u n+l=R.u”,
where L and R are the finitedifference operators
(L’u),=uj+~r(fi+1/2f;1/2),
(R.u)j=uj1(1r)(J;+,/2f,,/2).
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 CFLlike restriction
(2.13)
(2.14a)
(2.14b)
(2.15a)
(2.15b)
(2.16)
where aj + 1,2 is defined in (2.5). For a detailed proof of (2.15) and (2.16), see [3].
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 CFLlike 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 threepoint conservative TVD schemes of the form (2.12)
are generally firstorder accurate in space. When q = 1, the scheme is secondorder
accurate in time.
2.3. FirstOrder Accurate Backward Euler Implicit TVD Scheme
In this paper, we are only interested in efficient highresolution timedependent
methods for steadystate calculations. The backward Euler implicit TVD scheme is
the best choice in this oneparameter 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 firstorder accurate
unconditionally TVD scheme (2.12) with q = 1 into a secondorder accurate one.
Page 7
IMF’LICIT TVD SCHEMES
333
The backward Euler threepoint scheme in conservative form can be written as
ui”’ l+ n(f;;;,, f;$,, = 24;.
(2.17a)
For the purpose of this paper, the function Q(z) in (2.4) is chosen to be
(2.17b)
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 [2]. For convenience, we introduce the notation
C’(z) = $[Q(z) f z]
(2.18a)
and note that
C’(z)>0
for all z.
(2.18b)
Using (2.5) and (2.18a), we can rewrite the numerical fluxes f;.+ 1,2 in (2.4) as
~+~/~=fjC(~j+~/~)Aj+~/~u,
j;. 112 =fi C+ (aj 112) A, 112~.
(2.19a)
(2.19b)
It follows from (2.19) that (2.17) can be written in the form
u?+’  J.C(a~~~,,) A,, I,2~n+1 + K’(uJ’?~,,)
I
Aj 1,2~“+1 = u;.
(2.20)
Now, if 6 = 0 in (2.17b), then C’(z) = (lzl f z)/2, and (2.20) is a firstorder
accurate, upstream differencing, backward Euler implicit scheme. Equation (2.17)
differs from the upstream spatial differencing (with 6 =0) by the addition of a
numerical viscosity term with a coefftcient 6 > 0.
We show now that C’(z)30 implies that the scheme (2.17) is unconditionally
TVD (i.e., condition (2.10) is satisfied, independent of the value of A = At/Ax in
(2.17a).
To see that, we subtract (2.20) at j from (2.20) at j + 1 and get after rearranging
terms that
C1+~C~,/2+~Cj++1/21Aj+,/2Un+1
=Aj+ l/2 lln + lCj” l/ZAj
l/Z” n+l +K,~3/2Aj+3,2~n+?
(2.21a)
Here Cj$ 1,2 = C * (a;:$, ). Next we take the absolute value of (2.21a) and use
(2.18b) and the triangle inequality to obtain
Cl+~C&,/2+JC~,/,l lAj+1/2Unf11
G lAj+ I/ZU~I + ‘CjT 112 IAj 112~
n+ll +~C~3/21Aj+3/2Un+‘I. (2.21b)
Page 8
334
YEE, WARMING, AND HARTEN
Rearranging terms, we get
/A I+ 112 u”+‘I d Pi+,,, ‘“l+‘CC,~~/21’j+~/2~n+‘IC,~~/~IAj+~,~~nf’l]
‘CC~+,,,,IA,+,,Z~~~‘IC/+_~/~~~~~~,~U~+~~].
(2.21c)
That is,
IA,+ 1/2u ‘+‘I 6 IAj+l/2U”l +%(Ej+lZj),
(2.21d)
where
~j=CCi+1pIA,+1/2Un+‘I CJ+~,/~~A,~~/~U”+~I.
(2.21e)
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 SecondOrder Accurate Scheme
Next, we want to briefly review the design principle behind the construction of
secondorder accurate TVD schemes. This is a rather general technique to convert a
threepoint firstorder accurate (in space) TVD scheme (2.12) into a fivepoint
secondorder accurate (in both time and space, or just space) TVD scheme of the
same generic form. The design of highresolution 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 firstorder accurate scheme is TVD subject only to the CFLlike
restriction (2.16) independent of the particular form of the flux. Thus to achieve
secondorder accuracy while retaining the TVD property, we use the original TVD
scheme with an appropriately modified flux (f + g), i.e.,
,:+I +%(7r=,:,7r’:,,)=u,“,
(2.22a)
.~+1/2=~Cf;+fi+~+gj+gj+1Q(a,+1/2+~j+1/2)Aj+~/~~l, (222b)
where
Y,+ l/2 =
k,+ I  g,)lAj+ (/2u,
0,
Aj+ 1/2u + 0,
A ,+1,2u=o.
(2.22c)
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 secondorder 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
(2.22d)
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IMPLICIT TVD SCHEMES
335
with oj+ 1,2 = o(aj+ i,J and we choose
c(z) = Q(z) > 0
(2.22e)
for steadystate applications. It has the property that the steadystate solution is
independent of At. Or, we choose
CT(z) = i@(z) + AZ*) > 0 (2.22f)
for timeaccurate calculations. Note that if a(z) = (Q(z) + Lz*)/2, then (2.22) is
secondorder accurate in both time and space [3]. For transient calculations,
secondorder accurate in time is preferred.
The form of g in (2.22d) satisfies the relations [3]
gj=gt"jl, uj3 uj+lh
g(u, 6 u) = 0,
IYj+ l/21 = lgj+ l gjlll”j+ 1 #jl G d”j+ I/2),
(2.23a)
(2.23b)
g=Axo(a);+O((Ax)‘).
Relation (2.23a) shows that the modified numerical flux (2.22b) is consistent with
f(u). Relation (2.23b) shows that the meanvalue characteristic speed rj+ ,,z (2.22~)
induced by the flux g is uniformly bounded. Relation (2.23~) implies that (2.22b) is
secondorder 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
firstorder scheme (2.17). Therefore (2.22) is an upstream differencing scheme with
respect to the characteristic field (a + y). Moreover, we have the relation
+ yg:&);
i.e., C’ is now a function of (a + y)
sign(a + y) = sign(u)
(2.25)
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 secondorder accurate TVD
scheme depends on four points, i.e., yj+ 1,2 = T(Tcu~
is formally a livepoint scheme. We note, however, that
, , uj, uj+ 1, uj+ *), and thus (2.22)
lb, 4 4 WI = f(u)
(2.26)
for all v and w. Hence, for practical purposes, such as numerical boundary con
ditions, (2.22) can be regarded as essentially a threepoint scheme.
Page 10
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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
3”,+2.
(2.27)
In this case gj = g, + L = 0 in (2.22d), and thus the numerical flux (2.22b) becomes
identical to that of the original firstorder 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 secondorder accurate scheme to be
TVD, it has to have a mechanism that switches itself into a firstorder accurate
TVD scheme at points of extrema. Because of the above property, secondorder
accurate TVD schemes are genuinely nonlinear (i.e., they are nonlinear even in the
constant coefficient case).
Extension of the oneparameter family of threepoint TVD schemes (2.12) to
secondorder 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 firstorder accurate TVD scheme (2.12) into a
secondorder accurate one is closely related to the concept of artificial compression
[13, 143.
Truncation error analysis shows that the firstorder accurate scheme (2.12) is a
secondorder accurate approximation to solutions of the modified equation
(2.29)
where o(a) is defined in (2.22e) or (2.28). We note that the CFLlike restriction
(2.16) implies that o(a) 2 0; thus, the righthand side of (2.29) is a viscosity term.
Hence the firstorder 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 secondorder approximation to au/at + af/ax = 0 by applying the
firstorder scheme (2.12) to the modified flux (f+ g), where g is an approximation
to the righthand side of (2.29); i.e.,
g = dxa(a) g + O((fl~)~). (2.30)
The application of the firstorder scheme to (f + g) has the effect of canceling the
error due to the numerical viscosity to O((~X)~); thus g is an “antidiffusion” flux.
If we apply the firstorder 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 secondorder accurate TVD
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IMPLICIT TVD SCHEMES
337
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.,
with
gj=(l +COOj) gj,
o>o (2.31a)
(2.31b)
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 [2] for more details. From numerical experiments, o = 2 seems to be a
good choice.
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) thirdorder
monotonicity).
accurate (in regions of
2.6. Linearized Version of the implicit TVD Scheme
To solve for zJ+’ for the first or secondorder 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 steadystate 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 secondorder accurate implicit
TVD schemes. We will discuss the LNI for the secondorder accurate one. To get
the LNI for the firstorder accurate TVD scheme, we simply set g= y =0 in the
secondorder form.
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
[l ~C~(a+y)~+,,,dj+,,,+~C+(a+y)~~1,2dj~1,21(Un+1Un)
= x7;+ l/2 JI:
I/21>
(2.33a)
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YEE, WARMING, AND HARTEN
where the lefthand side equals
with d=u?+‘uU” and A.
I
7
I
E,dj1 +E,d,+E,dj+,=
I+ li2d=dit I d,. Rearranging terms, we get
~IIY~+~/~.T~~/~I,
with
(2.34a)
E, = K+(u+y)i”p1,2,
E,=~+IICC(U+~):+,,~+C+(,+~)~,,,I,
E, = AC (a + y);, ,,2.
(2.34b)
(2.34~)
(2.34d)
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 steadystate solution of (2.33) is
(i) consistent with the conservation form, and
(ii) a spatially secondorder accurate approximation to the steady state of
the partial differential equation
(iii) Independent of the timestep 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
secondorder method is a livepoint 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
timedependent solutions (transient solutions). However, it is a suitable scheme for
the calculation of steadystate solutions.
We can also obtain another TVD linearized form by setting y = 0 in (2.34), i.e.,
E,dj,+E,dj+E,dj+,=
JEfr+r/2JI,“l/J,
(2.35a)
with
E, = AC + (a; ,,*),
~2=l+w(u~+,,,)+C+(u;~,,,)],
Js = AC (a,“, 1,2).
(2.35b)
(2.3%)
(2.35d)
Scheme (2.35) is spatially firstorder accurate for the implicit operator and spatially
secondorder accurate for the explicit operator. It can be shown that (2.35) is still
TVD.
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IMPLICIT
Tk’D SCHEMES
339
3. GENERALIZATION TO ONEDIMENSIONAL
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 [15].
Now, we briefly describe the above approach of extending the secondorder
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
matrix
R(U) = (R’(U),..., R”(U))
(3.2a)
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
(3.2b)
W=R'U.
(3.3)
In the constant coefficient case, (3.1) decouples into m scalar equations for the
characteristic variables
awl ,aw/
e&+u z=o,
a’ = constant.
This offers a natural way of extending a scalar scheme to a constant coefficient
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340
YEE, WARMING, AND HARTEN
system by applying it “scalarly” to each of the m scalar characteristic equations
(3.4).
Let h1,2
denote some symmetric average of U, and U,, , (to be discussed
later); i.e.,
uj+l/2= yC"j3
u,+l).
(3.5)
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.,
Aj+ 112
U=Rj+1/2a/+1/2r Uj+,/2=Ri+',/2Alf,/2U.
(3.6)
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,
/= 1
(3.7b)
where
gf=S.maxCO, min(~~+,,21~~+,,21, S~O;~,,,CX~,,,)],
S = skn(uj+
,,2),
(3.7c)
and
I
k:+ I
0
L
Yj+1/2=
 s:Yu; + I/2>
a:+ l/2 # 09
a;, ,,2 = 0.
(3.7d)
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
with
$=(l +fde;,g;,
o’>O, (3.7e)
(y= I4+1,2~:h
I@+ I,21 + I+ I,21
(3.7f)
The w’ can be different from one characteristic field to another.
Similarly, we generalize the LNI form (2.33) to the system case by
[I‘(,< ,/>A,+ l/2 + iJ,T 1/2Aj l/21( U”+ ’  Un) = A[?+ l,2  q.. ,,2]
(3.8a)
or
ElDj,+E2Di+E:jDi+l= rl[F~‘+1/2$‘~,/2],
(3.8b)
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IMPLICIT TVD SCHEMES
341
with
(3.8~)
(3.8d)
(3.8e)
and
JjS
l/2 = R&
Dj= ,,“‘I  Uj’,
l/2 diag(C'(a'+ Y'):, ,,2)(R ~ 'I,"+ ,,2,
where the lefthand side of (3.8a) is equal to
(3.8f)
(3.W
Dj ‘Jjy 1/2Aj+ 1120 + ‘Jj? 1/2Aj I/zD,
with Aj+,,,2D=Dj,,Dj.
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 lefthand side.
Note that the total variation for the vector mesh function U of the constant coef
ficient case is defined as
(3.8h)
WV= f f
I4+1,21.
(3.9)
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
should satisfy
(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.
4. APPLICATIONS
INVISCID
TO ONEDIMENSIONAL
EQUATIONS
COMPRESSIBLE
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