Convergence of a Difference Scheme for Conservation Laws with a Discontinuous Flux.
Convergence is established for a scalar finite difference scheme, based on the Godunov or Engquist--Osher (EO) flux, for scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient. Other works in this direction have established convergence for methods employing the solution of  2  2 Riemann problems. The algorithm discussed here uses only scalar Riemann solvers. Satisfaction of a set of Kruzkov-type entropy inequalities is established for the limit solution, from which geometric entropy conditions follow. Assuming a piecewise constant coefficient, it is shown that these conditions imply L1-contractiveness for piecewise C1 solutions, thus extending a well-known theorem.
Introduction
. The subject of this paper is a nite dierence algorithm for
computing approximate solutions of the Cauchy problem for scalar conservation laws
of the form
1). The
ux k(x)f(u) has a possibly discontinuous spatial
dependence through the coe-cient k, which is allowed to have jump discontinuities.
A simple physical model corresponding to 1.1 is the Witham model of car tra-c
ow
on a highway [11]. The spatially varying coe-cient k corresponds to changing road
conditions. Equations of this type have been addressed by several authors in recent
years, often as the simplest examples of nonstrictly hyperbolic systems [6], [7], [20].
This approach is based on the observation that 1.1 can be written as a 2x2 system:
The eigenvalues associated with this system are  The system
fails to be strictly hyperbolic if f 0 can vanish, in which case the system is described
as resonant.
Even when k and the initial data u 0 are smooth, solutions to 1.1 develop discon-
tinuities, and so weak solutions are sought. A weak solution is a bounded measurable
function
Z
RR
Z
R
for all 2 C 1
(R  [0; 1)).
This paper considers the conservation law 1.1 within the framework of scalar
nite dierence schemes. The spatial domain R is divided into cells I
with centers at the points x Z. Similarly, the time
domain [0; 1) is discretized via t nt for resulting in time strips
I j;n be the characteristic function for the rectangle R n
October 30, 1999 jtowers@cts.com
J.D. Towers
The nite dierence scheme then generates, for each mesh size
constant approximation
The initial data is discretized via the piecewise constant approximation
x
Z
I j
where  j (x) is the characteristic function for the interval I j . The approximations u n
are generated by the explicit, conservation-form, time marching algorithm
Here, 1is based on a two point monotone (nondecreasing with
respect to the second argument, nonincreasing with respect to the rst) numerical
ux, h(v; u), consistent with the actual
ux, i.e., h(u; In order to maintain
the monotonicity of the scheme, the arguments of the numerical
ux are transposed
when the coe-cient k is negative, so that
In particular, this paper focuses on the closely-related Godunov [3] and Engquist-
Osher (EO) [2]
uxes. The Godunov
ux is derived by applying the exact solution
operator for 1.1, with constant to piecewise constant initial data. The numerica

ux that results is then h(v; Here u G (v; u) is the similarity
solution of the resulting Riemann problem with right and left states v and u, evaluated
anywhere along the vertical half line t > 0 in (x; t)-space where the jump in the
initial data occurs. The following formula for the Godunov
ux is derived in [14]:
min [u;v] f(w); if u  v
The Godunov
ux is Lipschitz continuous, and for f 2 C 1 it has piecewise continuous
partial derivatives. With the notational convention that a = min(a; 0),
it follows from 1.6 that
The EO
ux, which is dened by
is also Lipschitz continuous, but is smoother than the Godunov
ux. For f 2 C 1 , it
has continuous partial derivatives satisfying
(1.
Dierence Scheme for a Discontinuous Flux 3
It follows from 1.7 and 1.9 that kf 0 k1 serves as a Lipschitz constant for both
uxes.
In the case where the
ux f is convex or concave, with a single stagnation point,
the EO and Godunov
uxes are identical, except for the single case where the two
arguments are joined by a sonic shock, i.e., f 0 (v) < 0 < f 0 (u).
In the scheme 1.4, the coe-cient k is approximated at each cell boundary, resulting
in a discretized version of k,
where  j+ 1(x) is the characteristic function for the interval I j+ 1= [x j ; x j+1 ). The
discretized version k  has jumps at cell centers, as opposed to cell boundaries, i.e., the
discretization of k is staggered with respect to that of u. This results in a reduction
in complexity, compared with the approach where the two discretizations are aligned.
When the discretizations of the conserved quantity and
ux coe-cient are aligned,
more complicated 2x2 Riemann problems arise, and thus a 2x2 Riemann solver is
required. The concept of reducing complexity via mesh staggering is well known.
In [19], Temple used the Glimm scheme, solving 2x2 Riemann problems, to establish
existence and uniqueness for a 2x2 resonant system of conservation laws. The
singular function , used to prove compactness in x3 of this paper, originated in
[19], in a slightly dierent form. (Unlike the system 1.2, the system studied was not
reducible to a scalar conservation law, so the scalar scheme 1.4 discussed in this paper
is not applicable to that problem.) The authors of [12] and [13] studied the 2x2
Godunov method as it applies to
They adjoined the equation k treated 1.10 as a 2x2 system, for the purpose
of modeling a resonant system. Using to establish compactness, they proved that
the 2x2 Godunov method is convergent (modulo extraction of a subsequence) to a
solution. They established a bound on the total variation of a function that corresponds
to the function z  of this paper, but their method of analysis was dierent,
entailing a study of the various waves arising from the 2x2 Riemann problems. In
[13] it was observed that time independent bounds on derivatives (measured via )
had been achieved for the 2x2 Glimm and Godunov schemes, but not for any scalar
scheme that applies to 1.10. Section 3 of this paper provides a bound of this type
for equations of the form 1.1, which constitute a nontrivial subset of 1.10, and are
sometimes used to model the more general 1.10 [6], [7], [20] .
Outside of the dierence scheme methods, the front tracking approach has been
eective, computationally, and as an analytical tool for studying solutions to 1.1.
Reference [7] establishes existence, uniqueness, and asymptotic behavior of the solution
of 1.1, under essentially the same concavity assumptions about the
ux f as this
paper. The singular function , in the form appearing in x3 of this paper, is used
in [7] also. A bound on the variation, measured via , is achieved by studying the
wave interactions that arise. Additionally, satisfaction of a wave entropy condition
is established for the limit of the front tracking scheme, and uniqueness is shown to
follow from this entropy condition. In [6], it is shown that solutions of 1.1 and 1.10
depend continuously on k.
Reference [20] discusses computational di-culties that can arise with schemes for
1.1 and 1.10 that are based on the solution of 2x2 Riemann problems. Specically, the
4 J.D. Towers
random choice method and front tracking method can fail, due to variation blow-up,
and non-existence of solutions to 2x2 Riemann problems. The 2x2 Lax-Friedrichs
scheme does not experience these failures, as is shown in [20] by examples, since it
avoids solving Riemann problems. The scheme 1.4 discussed in this paper, which
is based on scalar Riemann solvers, is also unaected by these di-culties. In the
example provided in [20] where a solution to the Riemann problem fails to exist, the
source of this failure is a sign change in k. This provides the motivation for addressing
the case of indenite k in x1 and x2 of this paper.
This paper is organized as follows. Section 2 discusses the monotonicity and L 1 -
contractiveness properties of the scheme 1.4, including the case of indenite k. Section
3 focuses on the case of positive k, and establishes the main result of the paper,
convergence of a subsequence to a weak solution of the conservation law. Section
4 addresses the related issues of uniqueness and entropy satisfaction for the limit
solution.
2. Monotonicity. For the case of constant k, the theory of monotone schemes
has been essentially complete for many years [1], [5], [9], [17]. In that setting, approximations
generated by monotone schemes are well-known to share many of the
properties of the actual entropy weak solution to the conservation law 1.1, including
monotonicity, L 1 -contractiveness, and satisfaction of a discrete entropy inequality.
Their major drawback is that they are at best only rst order accurate even in regions
where the solution is smooth [5]. Nevertheless, they provide the starting point
for many of the modern higher order accurate schemes, some of which are constructed
by modifying the two-point monotone
ux function with higher order correction terms,
and then applying
ux limiters [4], [15]. The
ux limiters damp out spurious oscillations
that the correction terms often generate in regions of rapid transition.
A nite dierence scheme such as 1.4 is monotone if
When k is constant, monotonicity of the scheme follows from monotonicity of the
numerical
ux h under suitable CFL conditions. It follows that, in addition, the
computed approximations u n
remain within the convex hull of the initial data for all
n. The following proposition provides an analog of these properties for the variable k
situation.
Proposition 2.1. With the CFL condition 2kkk1 kf 0 k1  1 both the Godunov
and EO versions of the scheme are monotone. If k is nonnegative or nonpositive the
CFL condition can be relaxed to kkk1kf 0 k1  1. If the initial data u 0 (x) lies within
the
interval
j0 , for all j, and all n  0.
Proof. Let
. Expressing the three-point scheme as u n+1
the proof proceeds by showing that the partial derivatives @G=@u j+i are each non-
negative, from which monotonicity of the scheme follows. That the partial derivative
of G with respect to u j 1 is nonnegative is clear from
similar formula for @G=@u j+1 shows that it is also nonnegative. For @G=@u j , there
are four cases, depending on the signs of k j 1and k j+ 1. If k
Dierence Scheme for a Discontinuous Flux 5
The case where k j 1 0 and k j+ 1 0 is similar, and the calculation is omitted. For
the case where k j 1< 0 and k j+ 1> 0,
The case where k j 1> 0 and k j+ 1< 0 is similar and is omitted. The stated invari-
ance
of
with respect to the computed solution u
j follows from the monotonicity
of the scheme, along with the fact that f vanishes at u and u. Specically,
For the remainder of this paper it will be assumed that u
loc
denotes the space of locally integrable functions w
having bounded total variation, denoted TV (w). Also, it will be assumed that k is
constant for large x, specically, that there are constants k(+1), k(1), and X such
that . With the cell-average
type discretizations discussed in x1 for u 0 and k, the discretized versions also satisfy
these assumptions, and the various norms of the discretized quantities, u
are bounded uniformly in  by the corresponding norms for u 0 and k [1].
Proposition 2.2. With the CFL conditions in Proposition 2.1 and the assumption
that the initial data satises u
is constant for
large x, the scheme 1.4 is L 1 -contractive, i.e., the inequality
holds for a pair of approximate solutions u n
generated by the scheme. The
following inequality also holds:
Proof. Both solutions u n
remain in L 1
. The L 1 bound follows from
Proposition 2.1. The L 1 bound follows from the nite range of in
uence of the initial
data, along with the assumption that k is constant for large x. These observations
along with the fact that the scheme is monotone (due to the CFL condition) and
conservative, add up to the hypotheses of the Crandall-Tartar Lemma [1], giving
2.1. The inequality 2.2 follows from L 1 -contractiveness, applied inductively, to two
successive time steps, u n
and u n+1
j , of a single computed solution.
6 J.D. Towers
3. Convergence. In the case where the coe-cent k is constant, monotonicity
implies that the scheme is Total Variation Decreasing (TVD). However, even for
smooth, but nonconstant, k, the total variation of the solution to the conservation
law generally increases, at least initially, and so there is no hope that the numerical
scheme for variable k will be TVD.
As in [7], [12], and [19], the approach to convergence in this paper is to use the
singular mapping rst used by Temple in his study of a resonant hyperbolic system
[19]. Some assumptions about the
ux function f(u) will be required before dening
. The
ux function f 2 C 2 [0; 1] is assumed to be strictly concave on [0; 1], i.e.,
the interval (0; 1), f(u) > 0, with a
single maximum f  at u  2 (0; 1). These are similar to the conditions imposed on f
in [6] and [7]. Then the singular function is dened by
f
Z u
Here In the remainder of this paper, k will be
assumed to be bounded and strictly positive: 0 < k  k(x)  k. Then 1.5 becomes
The convergence result of this paper remains valid (with a more
restrictive CFL condition, and a larger bound on the variation of z  ) if k is allowed
to have nitely many points x where k(x )k(x In order to avoid obscuring
the main idea of the paper, the analysis of this more general case is not presented.
For each value of k in [k; k], (; is an increasing, 1 1
mapping. It is regular everywhere, except at the stagnation point u  , where both f 0
and @ =@u vanish. The following Lipschitz continuity relationships in u and k follow
directly from the denition of and the conditions imposed on the
kf 0 k1
The function maps a function w(x; t) into a new function z(x; t) via z(x;
(w(x; t); k(x)). The following elementary facts concerning z are readily veried.
Proposition 3.1. Let w
1. For each t  0, z(;
2. For each t  0, z(;
loc (R), and
R +B
It is now possible to state the main result of this paper.
Theorem 3.2. Let k 2 L 1
loc
taking constant values for large
x, and assume that 0 < k  k(x)  k. Let f 2 C 2 [0; 1], f 00   < 0,
in (0; 1), with a single maximum f at u  2 (0; 1). Let
Let the mesh size  ! 0 with  xed and satisfying the CFL
condition resulting in the sequence of approximations u  . Then,
there is a weak solution u of 1.1 and a subsequence u  i such that u
in L 1
loc (R  [0; 1)).
The rest of this section carries out the analysis required to prove Theorem 3.2.
The basic approach is standard as in [17] and [18], with the exception that the quantity
of immediate interest is z instead of the conserved variable u  . Compactness
for z  follows in the usual way from bounds on the total variation and the
along with a time-continuity estimate. Once the existence of a subsequential
limit z has been established, the invertibility of then allows the corresponding
Dierence Scheme for a Discontinuous Flux 7
solution u to be recovered from the limit z, with u  ! u guaranteed by the
continuity of . Finally, a version of the Lax-Wendro Theorem [10] demonstrates
that the limit u is a weak solution to 1.1.
Lemma 3.3. Let
For both the Godunov and EO versions of the scheme,
elsewhere.
Proof. Like (u; k),  is a strictly increasing function of u, so
assume that u j > u j+1 . For concave f , and u j  u j+1 , the Godunov
and EO
uxes are identical, so 1.9 holds for both
uxes. Also, for either
ux,
Z
h u
Z u j+1
Using the following decomposition of  j  j+1 ,
Z
Z
Z
it su-ces to show that the following two inequalities are satised:
Z u j+1
Z u j+1
For 3.3, if  are to the left of u  , so the integral
vanishes, and the inequality holds. So assume that
R u j+1
R
h u
R
h u In either case 3.3 holds, since then
Z
h u
Z u j+1
Z u j+1
Z u j+1
For the integral vanishes. If
Z u j+1
h u
Z u j+2
Z u j+1
h u
Z u j+1
8 J.D. Towers
and  denote forward and backward dierence operators. For example,
With the assumptions stated in
Theorem 3.2, uniformly, for all n  0, and all  > 0. Specically,
Proof. The superscript n is suppressed in the proof, except where two time levels
are of interest. Taking into account the jumps in z at the cell boundaries (due to jumps
in u  ) and the jumps at the cell centers (due to jumps in k  ), the total variation of
z(x) is
where  u
The second sum in 3.6, due to jumps in k  , is bounded by TV (k), using 3.1. For the
rst sum, let Summing over all of the jumps, and using the
fact that z(x) ! k(1) as x ! 1, results in
u
It follows from 3.7 that
Then,
The following identity will be useful in estimating
f
By Lemma 3.3,
Dierence Scheme for a Discontinuous Flux 9
The last inequality uses the fact that jh j+ 1j  f  , which follows directly from the
conditions on f , along with 1.6 and 1.8. Substituting this into 3.8, shifting indices,
and applying L 1 -contractiveness gives
f
f
The nal step in the proof is to estimate the term
j. For the jth term in
this sum,
Summing over j results in
Substituting this into the last inequality in 3.9 yields
When all of the terms are collected, the bound 3.5 stated in the theorem results.
Lemma 3.5. With the assumptions stated in Theorem 3.2, there is a constant L,
independent of the mesh size , such that for n > m  0,
Z
R
Proof. Taking into account the constant values of z  in each half-cell, the integral
on the left side of 3.11 is
An application of 3.2 and L 1 -contractiveness yields
Z
R
f
kf 0 k1
f  (n m)
J.D. Towers
The proof is completed by bounding
independently of , using 3.10.
The following is essentially the Lax-Wendro theorem [10].
Theorem 3.6. Let be a sequence of approximations
computed via the scheme 1.4 which converges in L 1
loc (R  [0; 1)) and with
uniformly bounded, to u 2 L 1
loc (R  [0; 1))
(R  [0; 1)). Then u is a weak
solution of 1.1.
Proof. Let
(R  [0; 1)) and  t n
.
Jx such
jxj  B. As in [10], multiply the scheme 1.4 by
n  0, then sum by parts to get
where h j+ 1= h(u  arguments [11], the sums
involving
j converge to their integral counterparts in 1.3, as
It remains to show that
xt
Z
RR
kf(u) x dxdt:
The proof of 3.12 reduces to verifying that
Z
I j
tx
The sum appearing in 3.13 does not exceed
Z
R
xZ
R
which proves 3.13. The expression on the left side of 3.14 is bounded by
Z
The rst integral in 3.15 tends to zero by assumption, and the following estimate
tx
Z
tx
Dierence Scheme for a Discontinuous Flux 11
shows that the second integral also approaches zero.
It is now possible to prove Theorem 3.2.
Proof. The CFL condition guarantees that the computed solutions u  remain
within [0; 1]. An application of Proposition 3.1 gives uniform (with respect to both
t and ) bounds on kz  (; t)k 1 and kz  (; t)k compact interval
Theorem 3.4 provides a uniform bound on TV (z  (; t)). Finally, from
Lemma 3.5, it follows that
where the constant C is independent of  and t. By standard compactness arguments
applied to the sequence z  , there is a subsequence, z  i , which converges in
loc (R  [0; 1)) to some function z 2 L 1
loc (R  [0; 1))
(R  [0; 1)). There is
a further subsequence, also denoted z  i , which converges a.e. to z. Let u(x;
Due to the strict monotonicity of (; k), the function u is well-
dened a.e.,
loc (R  [0; 1))
(R  [0; 1)). Dropping
the subscript on , it remains to show that u  ! u a.e. and in L 1
loc (R  [0; 1)).
Using the fact that u
ju u  jdxdt
The rst integral tends to zero by the bounded convergence theorem, due to the
continuity of 1 as a function of its rst argument. For the second integral, an
estimate of j 1 (z  ; dierentiation gives
@k
@k
R u
When the numerator and denominator are each expanded in a Taylor series about u  ,
the result is
@ 1
@k
2k
kf 00 k1
ju u  j:
With this estimate, convergence of the second integral follows from the fact that
Having established convergence in L 1
loc (R  [0; 1)), there is
yet a further subsequence u  , which converges to u a.e., as well as in L 1
loc (R[0; 1)).
Theorem 3.6 proves that u is a weak solution of the conservation law.
4. Entropy satisfaction. Even for constant k, solutions of 1.1 are not necessarily
unique. Additional conditions, usually referred to as entropy conditions, are
required to single out the physically relevant solution. When k 2 C 1 the Kruzkov
entropy condition applies [8]. Specically, uniqueness is guaranteed if
Z
RR
dxdt  0
holds for all c 2 R, and for all nonnegative 2 C 1
0 (R R
condition 4.1 is not directly applicable if k is discontinuous, as was observed in [7].
12 J.D. Towers
For k discontinuous, the authors of [7] proved uniqueness within the class of solutions
that satisfy a wave entropy condition. The wave entropy approach has the advantage
of not requiring that the solution satisfy additional regularity conditions.
The approach in this section is to concentrate on the Godunov version of the
scheme, and assume, as in [6] and [7], that k has nitely many jumps. Proposition 4.1
ensures that the limit solution u satises the Kruzkov entropy inequalities 4.1 locally,
away from the jumps in k. Proposition 4.3 provides Kruzkov-type entropy inequalities
that apply when the test function has support which intersects one or more jumps in k.
Theorem 4.5 shows that these entropy inequalities imply geometric entropy conditions
for piecewise C 1 solutions, and Theorem 4.6 applies the geometric entropy conditions
to show uniqueness with the additional assumption that k is piecewise constant.
Proposition 4.1. In addition to the conditions stated in Theorem 3.2, let k
be piecewise C 1 , with a bounded derivative, jk 0 (x)j   for all x, and with nitely
many jumps (in k and k 0 ) , located at  1 <  . Let u  be a convergent
subsequence generated by the scheme 1.4 using the Godunov
ux, converging to u,
as in Theorem 3.2. Then the Kruzkov entropy inequalities 4.1 hold for every real
number c, and every smooth test function  0 with compact support in t > 0,
g.
To avoid complications arising from the discontinuity in V 0 the
following entropy inequality is established for smooth V before proceeding with the
proof of Proposition 4.1.
Lemma 4.2. In addition to the assumptions of Proposition 4.1, let (V; F ) be
a convex entropy pair for 1.1, i.e., V is convex and F assume that
1]. For every smooth test function  0 with compact support in t > 0,
g, and every c 2 R, the following inequality holds:
Z
RR
Z
RR
Proof. Write the scheme 1.4 as
Let u G
the Godunov discrete entropy
ux dened by H j+ 1=
j+ 1), the discrete entropy inequality
holds for w n
. After rearranging terms, a discrete entropy inequality for the scheme
results:
Multiplying by a smooth, nonnegative test function with compact support in t > 0,
g, and proceeding as in the proof of Theorem 3.6 gives
xt
xt
Dierence Scheme for a Discontinuous Flux 13
As  approaches zero, the rst sum in the top line of 4.2 converges to
in the proof of the Lax-Wendro theorem. For the second sum in the rst line of 4.2,
convergence to
reduces to establishing
tx
where J , N , B, and T , are dened as in the proof of Theorem 3.6. Using the fact
that kV 0 k1 kf 0 k1 is a Lipschitz constant for H j+ 1, the expression on the left side of
4.3 is bounded by
Z u G
The rst term in 4.4 tends to zero. Using jF and the fact that u G
1lies
between
and u n
j+1 , the second term does not exceed
Z
The estimate 3.16 proves that this term tends to zero also. Consider the second line
of 4.2. Since k 2 C 1 within the support of ,
xt
Z
RR
The last term in 4.2 to be dealt with is xt
))=t. Expanding
the divided dierence in a Taylor series yields
Due to L 1 -contractiveness, and the time continuity estimate 3.10, the sums involving
the last two terms on the right side of 4.5 approach zero as  ! 0. The rst term on
the right side of 4.5 approaches V 0 (u)f(u)k 0 in L 1
loc (R R + ), which gives
xt
Z
RR
It is now possible to prove Proposition 4.1.
Proof. As in [8], approximate the convex function ju cj by a sequence of twice
continuously dierentiable convex functions V i . Apply Lemma 4.2 for each V i , and
14 J.D. Towers
Let -(x  i ) denote the Dirac measure with support located at  i .
Proposition 4.3. With the same assumptions as in Proposition 4.1, the following
inequality holds for the special case where c = u  , for all nonnegative 2
0 (R R
Z
RR
ju
Z
RR
dxdt  0:
Proof. Proceeding as in the proof of Lemma 4.2 gives an inequality of the form
4.2, with V and the corresponding versions of F (u) and H j 1. The
sums in the rst line of 4.2 converge to their integral counterparts, as in the proof of
Lemma 4.2. For the second line of 4.2, there are  n
such that
Using the fact that h j 1= f(u G
With this inequality, 4.2 becomes
xt
xt
The term in the bottom line of 4.7 converges to the integral in the second line of 4.6.
This can be veried by breaking the spatial portion of the sum in the bottom line of
4.7 into sums over intervals where k is dierentiable, and isolating the nite number
of cells where the jumps in k are concentrated.
Let u be a piecewise smooth solution to 1.1. It follows from a standard test
function calculation that the Rankine-Hugoniot condition across a jump in k at one
of the points
where the subscripts L and R refer to limits from the left and right, respectively, at
the jump in k.
Lemma 4.4. Let F )). For a pair of states uL , uR
satisfying the Rankine Hugoniot condition 4.8,
kRF
Proof. Take kL  kR ; the other case is similar. By considering the various cases
for a pair of states (u L ; uR ) which satisfy 4.8, the following relationships result:
Dierence Scheme for a Discontinuous Flux 15
1. uR > u  > uL ) kRF
2.
3.
4. uR < u  < uL ) kRF
The rst three cases cover the situation where the equation on the right side of 4.9 is
satised. In each of those cases, the inequality on the left side of 4.9 also holds. Case
4 is the only case where the right side of 4.9 fails. In that situation, the left side of
4.9 also fails, since uL > u  ) fL < f  , and so
The entropy inequalities 4.1 and 4.6 yield the following fact concerning the satisfaction
of geometric entropy conditions.
Theorem 4.5. In addition to the assumptions of Proposition 4.3, assume that
the limit solution u is piecewise C 1 . Suppose that k jumps from kL to kR at
and u is C 1 in some neighborhood of the point each side of
the notation
For a discontinuity located at away from the jumps in k, assume that u is
some neighborhood of each side of a C 1 curve
(t). With
the following standard entropy condition holds:
where the shock speed
Proof. For 4.10, let N( a neighborhood of the type described in the
statement of the theorem. Let   0 be a smooth test function with support in the
rectangle
centered at extending backward and forward
in time from t 0 by an amount  . Assume that  and  are small enough
that
standard test function calculation, applied to 4.6, gives
Z
The integral over
can be made arbitrarily small by shrinking the width  of the rectangle. It follows
that kRF j. The entropy condition 4.10 then follows from
Lemma 4.4. The geometric entropy condition 4.11 follows directly from Proposition
4.1. For k 2 C 1 , it is well known that if 4.1 holds for all c 2 R, then the geometric
entropy condition 4.11 is satised [8].
The geometric entropy condition 4.11 is the usual geometric entropy condition
satised by shocks in the smooth k case. It requires that characteristics on both sides
of the shock extend toward the x-axis when followed backward in time from the shock.
The geometric entropy condition 4.10 requires that the characteristics on at least one
side extend toward the x-axis.
In the constant-k setting, the author of [16] established L 1 -contractiveness of
solutions which satisfy the geometric entropy condition 4.11. The following
theorem is presented as evidence that limits of the scheme 1.4 are the physically
J.D. Towers
relevant solutions. It extends the theorem in [16] to the case of piecewise constant k,
assuming that the additional geometric entropy condition 4.10 is satised.
Theorem 4.6. In addition to the previous assumptions, let k be piecewise con-
stant, with nitely many jumps, located at  1 <  Let u and v be
solutions of 1.1 satisfying the geometric entropy conditions 4.10
and 4.11. Assume that the initial data u 0 and v 0 are piecewise C 1 , and u
Then, for t  0,
Proof. Following [16], the approach is to show that the time derivative of the
integral on the left side of 4.12 is nonpositive. That integral is broken up into integrals
over segments where u v does not change sign. This decomposition yields
which is constant within the interval Dierentiating
4.13 with respect to time results in terms of the form
d
dt
_
In [16], it is noted that 4.14 holds even if there are shocks in the interior of
For essentially the same reason, i.e., the Rankine-Hugoniot condition 4.8, the relationship
4.14 holds even if one or more of the points  j lie within
(or x i+1 ) away from the jumps in k, the contribution to 4.14 from x i (or x i+1 ) is non-
positive, since the argument in [16] applies in this case, due to the geometric entropy
condition 4.11. The only case remaining is where u v changes sign at a jump in k,
say x
assume that kL < kR , the other case
being similar. There are contributions to 4.13 from two consecutive terms of the form
4.14. Taking into account that _
along the line the total contribution is
The situation where  can be eliminated, since then the Rankine-Hugoniot
condition 4.8 implies that both  i and  i+1 vanish. Take the case where
1. The other case is
similar. Then 4.15 becomes
In order for there to be a sign change, at least one of uR < uL , v R > v L must hold.
Take the case where uR < uL . With kL < kR and uR < uL , the geometric entropy
condition 4.10 and Rankine-Hugoniot condition 4.8 require that uL < u  . This is
easily veried by viewing being determined by
the intersection of a horizontal line with the graphs of kL f(u) and kR f(u). Since
vL < uL  u  , and f is increasing in (0; u  ), it is clear that f(vL ) < f(uL ). Then
using 4.8, the expression 4.16 is equal to
Dierence Scheme for a Discontinuous Flux 17
Next, take the case where v R > v L . With kL < kR and v R > v L , the Rankine-Hugoniot
condition 4.8 requires that v R > u  . If v L  u  , then uL > v L ) uL > u  .
The entropy condition 4.10 then requires that uR  u  . Then, with 4.8, uL > v L )
uR > v R , which is a contradiction, so it must be that v L < u  . If also uL < u  , then
case 4.16 reduces to
If uL > u  , the entropy condition 4.10 requires that uR  u  . Then, uR <
In this case 4.16 equals
2kR



--R

Monotone Di
One sided di

On a class of high resolution total-variation-stable nite dierence schemes

Stability of conservation laws with discontinuous coe-cients
Convex conservation laws with discontinuous coe-cients
First order quasilinear equations in several independent variables
Accuracy of some approximate methods for computing the weak solutions of a
Systems of conservation laws
Numerical methods for conservation laws
A comparison of convergence rates for Godunov's method and Glimm's method in resonant nonlinear systems of conservation laws
Suppression of oscillations in Godunov's method for a resonant non-strictly hyperbolic system
the entropy condition
High resolution schemes and the entropy condition
An example of an L 1
On convergence of monotone di
Shock waves and reaction-diusion equations
Global solution of the cauchy problem for a class of 2x2 nonstrictly hyperbolic conservation laws
The solution of nonstrictly hyperbolic conservation laws may be hard to compute
--TR

--CTR
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Adimurthi , Siddhartha Mishra , G. D. Veerappa Gowda, Conservation law with the flux function discontinuous in the space variable-II, Journal of Computational and Applied Mathematics, v.203 n.2, p.310-344, June, 2007
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conservation laws;difference approximations;discontinuous coefficients