A version of the Glimm method based on generalized Riemann problems

0.39 MB·English

by  John M. Hong

A version of the Glimm method
based on generalized Riemann problems
7
0
0
2
n May 17, 2006
a
J

2 John M. Hong1 and Philippe G. LeFloch2

]
P
A
Abstract
.
h
t We introduce a generalization of Glimm’s random choice method, which provides us with
a
an approximation of entropy solutions to quasilinear hyperbolic system of balance laws.
m
Theflux-functionandthesourcetermoftheequationsmaydependontheunknownaswell
[ as on the time and space variables. The method is based on local approximate solutions

of thegeneralized Riemann problem, which form building blocks in our scheme and allow
1
v ustotakeintoaccount naturallytheeffectsofthefluxandsourceterms. Toestablish the
3 nonlinear stability of these approximations, we investigate nonlinear interactions between
6 generalizedwavepatterns. Thisanalysisleadsustoaglobalexistenceresultforquasilinear
0 hyperbolic systems with source-term, and applies, for instance, to the compressible Euler
1 equationsingeneralgeometriesandtohyperbolicsystemsposedonaLorentzianmanifold.
0
7
0 1. Introduction
/
h
t 1.1 Hyperbolic systems of balance laws. This paper3 is concerned with the approxi-
a
mation of entropy solutions to the Cauchy problem for a quasilinear hyperbolic system
m
v: ∂tu+∂xf(t,x,u)=g(t,x,u), t>0, x∈R, (1.1)
i
X
r u(0,x)=u0(x), x∈R, (1.2)
a
where u = u(t,x) ∈ Rp is the unknown. We propose here a generalized version of the Glimm
scheme[10]whichallowsustodealwithalargeclassofmappingsf,g andtakeintoaccountthe
geometriceffectofthefluxandsourceterms. Ourschemeisbasedonanapproximatesolverfor
the generalized Riemann problem, based on an asymptotic expansion introduced by LeFloch
and Raviart [17]. The approach provides high accuracy and stability, under mild restrictions
on the equation and the data.
In (1.1), the flux f = f(t,x,u) ∈ Rp and the source-term g = g(t,x,u) ∈ Rp are given
smooth maps defined for all (t,x,u) ∈ R ×R×U, where U is a small neighborhood of the
+
1DepartmentofMathematics,National CentralUniversity,Chung-Li32054, Taiwan.
E-mail: [email protected]
2LaboratoireJ.-L.Lions&C.N.R.S.,UniversityofParisVI,75252Paris,France.
E-mail: lefl[email protected]
3This paper is based on notes written by the second author in July 1990, for his Habilitation memoir
(Chapter XI)attheUniversityofParisVI.
1
origin in Rp, and the initial data u : R → U is a function with bounded total variation. We
0
assumethattheJacobianmatrixA(t,x,u):= Df(t,x,u)admitsprealanddistincteigenvalues,
Du
λ (t,x,u)<λ (t,x,u)<...<λ (t,x,u),
1 2 p
and therefore a basis of right-eigenvectors r (t,x,u) (1 ≤ j ≤ p), Finally, we assume that
j
each characteristic field is either genuinely nonlinear (∇λ (t,x,u)·r (t,x,u) 6= 0) or linearly
j i
degenerate (∇λ (t,x,u)·r (t,x,u)=0).
j i
One importantmotivationforconsideringgeneralbalancelaws(1.1) comesfromthe theory
of generalrelativity. In this context, the vector u typically consists of fluid variables as well as
(first order derivatives) of the coefficients of an unknown, Lorentzian metric tensor. (See [3, 5]
and the reference therein.) One can also freeze the metric coefficients and concentrate on the
dynamics of the fluid. For instance, the compressible Euler equations describing the dynamics
of a gas flow in general geometry read:
∂ a ∂ a
x t
∂ ρ+∂ (ρv)=− ρv− ρ,
t x
a a
∂ a ∂ a
∂ (ρv)+∂ (ρv2+p)=− x (ρv2)− t ρv, (1.3)
t x
a a
∂ a ∂ a
x t
∂ (ρE)+∂ (ρvE+pv)=− (ρvE+pv)− ρE
t x
a a
where a=a(t,x)>0 can be regardedas the cross section of a time-dependent (moving) duct,
and ρ, v, p(ρ,e), e, and E = e+u2/2 are the density, velocity, pressure, internal energy, and
totalenergyofthegas,respectively. Thesystem(1.3)describesasituationwherethefluiddoes
notaffectthe variationofthe duct; i.e.the function a(t,x) is givenand,for simplicity, smooth.
Thesystem(1.3)isoftheform(1.1)withu=(ρ,ρv,ρE)T, f =f(u)=(ρv,ρv2+p,ρvE+pv)T
and g =g(t,x,u)=−∂xag (u)− ∂tau where g (u)=(ρv,ρv2,ρvE+pv)T.
a 1 a 1
We are interested in solutions to (1.1)-(1.2) which have bounded total variation in space
for all times and satisfy the equations in the sense of distributions, together with an entropy
condition [15, 8, 16]. In the special case that
f =f(u), g =0,
the existence of global entropy solutions was established by Glimm [10], assuming that the
initial data u (x) has sufficiently small total variation. Recall that two main ingredients in
0
Glimm’srandomchoicemethodare(1)thesolutionsofRiemannproblemsand(2)aprojection
step based on a sequence of randomly chosen points.
Let us first indicate some of the earlier work on the subject. The system (1.1) with
f =f(x,u), g =g(x,u),
was treated in pioneering work by Liu [20, 21], via a suitable extension of the Glimm method:
the approximate solutions are defined by pasting together steady state solutions, i.e., solutions
v =v(x) of the ordinary differential equation
d
f(x,v) =g(x,v).
dx
(cid:0) (cid:1)
He established the existence of solutions defined in a finite interval of time [0,T) as long as
either T or the L1 norms of g and ∂g/∂u are sufficiently small. Next, assuming in addition
that the eigenvalues of the matrix A(x,u) never vanish (so that no resonance takes place), Liu
deduced a globalexistence result (with T =+∞). Steady-state solutions were also used in the
work by Glimm, Marshall, and Plohr [12].
2
Formoregeneralmappingsf,g,the existencefor(1.1)-(1.2)isestablishedbyDafermosand
Hsiao [7] and Dafermos [8, 9]. They assume that f (u∗,t,x) = g(u∗,t,x) = 0 at some (equi-
x
librium) constant state u∗, hence u∗ is a solution of (1.1) around which (1.1) can be formally
linearized. They also require that the linearized system satisfies a dissipative property. Their
main result concerns the consistency and stability of a generalization of the Glimm method,
yielding therefore the global existence of entropy solutions to (1.1). In [7], the approximate
solutions to the Cauchy problem on each time step are based on classical Riemann solutions
with initial data suitably modified by both the source term g and the map θ :=A−1f .
x
Next, Amadori et al. [1, 2] developed further techniques to establish the existence of solu-
tionsforalargeclassofsystemshavingf =f(u)andg =g(x,u),anddiscussedDafermos-Hsiao
dissipative condition. For some particular systems (of two or three equations) the condition
thatthetotalvariationbesmallcanberelaxed;seeforinstanceLuskinandTemple[22],Groah
andTemple[11],Barnes,LeFloch,Schmidt,andStewart[3],andthereferencescitedtherein. In
thesepapers,thedecreasingofatotalvariationfunctional(measuredwithrespecttoasuitable
chosen coordinate) was the key to establish the stability of the scheme.
1.2 A new version of the Glimm method. In the presentpaper we provideanalterna-
tive approach to Dafermos-Hsiao’s method, and introduce a generalized version of the Glimm
scheme for general mappings f,g. Integrability assumptions will be required (and discussed
later on) on the matrix A and the mapping q :R ×R×U →Rp defined by
+
∂f
q(t,x,u):=g(t,x,u)− (t,x,u). (1.4)
∂x
Itshouldbeemphasizedthatonlythiscombinationofthesourceandthefluxwillbeimportant
in our approach, which can be summarized as follows.
First, we study the generalized Riemann problem associated with the system (1.1), i.e. the
Cauchyproblemwith piecewise constantinitial data. The existenceof solutionsdefined locally
in spacetime in a neighborhood of the initial discontinuity was studied in Li and Yu [18] and
Harabetian [13]. Contrary to the case where f,g only depend upon the unknown u, no closed
formula is available for the solutions of the generalizedRiemann problem. We propose here an
approximate Riemann solver, inspired by a technique of asymptotic expansion introduced by
Ben-ArtziandFalcovitz[4](for the gasdynamics equations)andLeFlochandRaviart[17](for
general hyperbolic systems of balance laws); see also [6].
OurschemeforsolvingapproximatelythegeneralizedRiemannproblemcanbere-interpreted
as a splitting algorithm (the hyperbolic operator and the source term being decoupled). Since
an approximate (rather than an exact) solution to the generalized Riemann problem is used,
it is crucial to establish an error estimate which we achieve in Proposition 2.1 below, under a
mild assumptionon the data u ,f,g. This estimate willbe necessaryto ensure the consistency
0
of our generalized Glimm method.
Second, we study the nonlinear interaction of waves issuing from two generalized Riemann
problems, and establish a suitable extension of Glimm’s estimates [10] to the general system
(1.1); cf. Proposition 3.3. This is a key, technical part of our analysis.
Third,weintroduceourschemeandproveitsstabilityintotalvariation,underthe assump-
tionthattheinitialdatau hassufficientlysmalltotalvariationandthatthetotalamplification
0
dueto(thederivativesof)f,g tothetotalvariationofthesolutionissufficientlysmall;cf.The-
orem 4.3. More precisely, we impose that
∂2A ∂2A ∂q
, , q,
∂t∂u ∂x∂u ∂u
are sufficiently small in L1(R ×R).
+
Finally, we conclude with the convergenceof the proposedscheme (Cf. Theorem5.1)which
yieldstheglobalexistenceofentropysolutionsfortheCauchyproblem(1.1)-(1.2). Thesolution
3
satisfies an entropy inequality and has bounded total variation in x for all t ≥ 0. Our results
cover in particular the case
∂ u+∂ f(u)=g(t), (1.5)
t x
for which global existence of entropy solutions is established under the sole assumption
+∞
|g(t)|dt<<1. (1.6)
Z0
Without further restriction on the flux f, this condition is clearly necessary in order to apply
the Glimm method, since, for instance in the trivial case p=1 and f =0, (1.5) reduces to the
differential equation
∂ u=g(t). (1.7)
t
Ononehand,the condition(1.6)holdsifandonlyifeverysolutionof (1.7)remainscloseto
a constant state, which is a necessary condition in order to apply the Glimm method. On the
otherhand,whenoneoftheeigenvaluesofthesystem(1.1)vanishes,theamplitudeofsolutions
couldbecome arbitrarilylargeand the solutions wouldnotremainbounded —exceptwhenthe
source term satisfies a “damping” property in time.
Asadirectapplication,theglobalexistenceofentropysolutionsto(1.3)follows,ifthesource
g and its derivative ∂g are sufficiently small in L1(R ×R), which is the case, for instance, if
∂u +
the support of (a ,a ) is sufficiently small.
t x
2. An approximate solver for the generalized Riemann problem
In the present section, we introduce an approximate solution to the generalized Riemann
problemassociatedwiththesystem(1.1),andwederiveanerrorestimates(seeProposition2.1
below).
Given t > 0, x ∈ R, and two constant states u ,u ∈ Rp, we consider the generalized
0 0 L R
Riemannproblem,denotedbyR (u ,u ;t ,x ), andconsistingofthe followingequationsand
G L R 0 0
initial conditions:
∂ u+∂ f(t,x,u)=g(t,x,u), t>t , x∈R, (2.1)
t x 0
u , x<x ,
u(0,x)= L 0 (2.2)
u , x>x .
R 0
(cid:26)
Replacingf andg in(2.1)byf(t ,x ,u)and0,respectively,theproblemR (u ,u ;t ,x )
0 0 G L R 0 0
reduces to the classical Riemann problem,which we denote by R (u ,u ;t ,x ), that is the
C L R 0 0
equations
∂ u+∂ f(t ,x ,u)=0, u(t,x)∈Rp, t>t , x∈R (2.3)
t x 0 0 0
togetherwiththeinitialdata(2.2). ThisproblemwassolvedbyLaxundertheassumptionthat
the initialjump |u −u | be sufficiently small: the solutionto R (u ,u ;t ,x ) is self-similar
R L C L R 0 0
(i.e.dependsonlyon x−x0)andconsistsofatmost(p+1)constantstatesu =u ,u ,...,u =
t−t0 L 0 1 p
u , separated by rarefaction waves, shock waves or contact discontinuities; see Figure 2.1.
R
The following terminology and notation will be used throughout this paper. Let W =
C
W (ξ;u ,u ;t ,x ) be the solution of R (u ,u ;t ,x ) with ξ = (x−x )/(t−t ). We say
C L R 0 0 C L R 0 0 0 0
that the problem R (u ,u ;t ,x ) is solved by the elementary waves (u ,u ) (i = 1,...,p)
C L R 0 0 i−1 i
ifeachconstantstate u belongsto the i-wavecurveW (u )issuedfromthe stateu inthe
i i i−1 i−1
phasespace,and(u ,u )iscalledani−waveofR (u ,u ;t ,x ). Whenthei-characteristic
i−1 i C L R 0 0
4
field is genuinely nonlinear, the curve W (u ) consists of two parts, the i-rarefaction curve
i i−1
andthei-shockcurveissuingfromu ;ifi-characteristicfieldislinearlydegenerate,the curve
i−1
W (u )is a C2 curveofi-contactdiscontinuities. Callε the strengthofthei-wave(u ,u )
i i−1 i i−1 i
along the i-curve, so that, for a genuinely nonlinear i-field, we can assume that ε ≥ 0 if
i
(u ,u ) is a rarefaction wave, and ε ≤0 if (u ,u ) is a shock wave. On the other hand, ε
i−1 i i i−1 i i
has no specific sign if (u ,u ) is a contact discontinuity.
i−1 i
Letε (u ,u ;t ,x )denotethewavestrengthofthei-wave(u ,u )intheRiemannprob-
i L R 0 0 i−1 i
lemR (u ,u ;t ,x ),andvectorε=(ε ,...,ε )denotethewavestrengthofR (u ,u ;t ,x )
C L R 0 0 1 p C L R 0 0
(so |ε| is equivalent to the total variation of W (ξ;u ,u ;t ,x )). In addition, we let σ− =
C L R 0 0 i
λ (u ,t ,x ) and σ+ =λ (u ,t ,x ) be the lower and upper speeds of the i-rarefaction wave
i i−1 0 0 i i i 0 0
(u ,u ) respectively, and σ be the speed of the i-shock or i-contact discontinuity. If the
i−1 i i
i-wave is a shock or a contact discontinuity we set σ− =σ+ =σ .
i i i
From the implicit function theorem we deduce that the states u and the speeds σ± are
i i
smoothfunctions of u , u , t , andx . Moreover,one cancheckthat u =u +O(1)|u −u |
L R 0 0 i L R L
(i=0,1,...,p), and, for an i-shock (u ,u ),
i−1 i
σ =λ (u ;t ,x )+O(1)|u −u |, i=1,2,...,p,
i i i−1 0 0 i i−1
where O(1) is bounded function possibly depending on u ,u ∈U, t ≥0, and x ∈R.
L R 0 0
Consider next the generalized Riemann problem on which a large literature is available
[18,13,4,6,17]. First,werecall[18]thatthesolutionofR (u ,u ;t ,x )ispiecewisesmooth
G L R 0 0
andhasalocalstructurewhichissimilartotheoneoftheassociatedclassicalRiemannproblem
R (u ,u ;t ,x ). Following[17]weconsideranapproximateRiemannsolutionoftheproblem
C L R 0 0
R (u ,u ;t ,x ), denoted by W (t,x;u ,u ;t ,x ) and defined by
G L R 0 0 G L R 0 0
W (t,x;u ,u ;t ,x )=W (ξ)+(t−t )q(t ,x ,W (ξ)) (2.4)
G L R 0 0 C 0 0 0 C
for t>t and x∈R. Here, the function q(t,x,u) is given by (1.4), and
0
x−x
0
ξ = , W (ξ)=W (ξ;u ,u ;t ,x ).
C C L R 0 0
t−t
0
Observe that the function W (t,x;u ,u ;t ,x ) is constructed as a superposition of the cor-
G L R 0 0
responding classical Riemann solution W (ξ;u ,u ;t ,x ) and an asymptotic expansion term
C L R 0 0
(t−t )q(t ,x ,W (ξ)) (see Figure 2.2).
0 0 0 C
Within a regionwhere function W (ξ) is a constant, the function W (t,x;u ,u ;t ,x ) is
C G L R 0 0
a linear function of t, namely,
x
W (t,x;u ,u ;t ,x )=u +(t−t )q(t ,x ,u ), σ+ < <σ− (2.5)
G L R 0 0 i 0 0 0 i i t i+1
for i = 0,1,...,p. By convention, σ+ := −∞ and σ− := +∞. Whenever there will be no
0 p+1
ambiguity, we will use the notation W (t,x) or W (t,x;u ,u ) for W (t,x;u ,u ;t ,x ).
G G L R G L R 0 0
To describe the structure of W (t,x;u ,u ;t ,x ), it is convenientto say that the approx-
G L R 0 0
imate solution W (t,x,u ,u ;t ,x ) consists of an i-wave (u ,u ) if (u ,u ) is an i-wave
G L R 0 0 i−1 i i−1 i
of the corresponding classical Riemann solution W (ξ;u ,u ;t ,x ).
C L R 0 0
5
Figure 2.1 : Classical Riemann solution (p=2),
u , u and u are constant states, u=u(ξ) is a function of ξ = x.
L 1 R t
Figure 2.2 : Generalized Riemann solution (p=2),
u (t), u (t), u (t) are functions of t and u˜(t,ξ) is constructed by (2.4).
L 1 R
We nowprovethatthe function W (t,x) defined in (2.4) approximatelysolvesthe problem
G
R (u ,u ;t ,x ), by evaluating the discrepancy between W (t,x) and the exact solution of
G L R 0 0 G
R (u ,u ;t ,x ). Given any s > 0 and r > 0, and any C1 function θ : R ×R → R with
G L R 0 0 +
compact support, we now show that the term
t0+s x0+r
∆(s,r;θ):= {W ∂ θ+f(t,x,W )∂ θ+g(t,x,W )θ dxdt (2.6)
G t G x G
Zt0 Zx0−r (cid:17)
is of third order in r,s, provided that the condition (2.7) holds.
Proposition 2.1. Let θ : R ×R → R be a compactly supported, C1 function. Then, for
+
every (t ,x ) ∈ R ×R, u ,u ∈ U, and any positive numbers s,r satisfying the (Courant-
0 0 + L R
Friedrichs-Levy -type) stability condition
s
sup|λ (t,x,u)|≤1 (2.7)
i
r
(the supremum being taken over 1≤i≤p, (t,x)∈R ×R, and u ∈U), the
+
6
function W (t,x)=W (t,x;u ,u ;t ,x ) satisfies
G G L R 0 0
x0+r x0+r
∆(s,r;θ) = W (t +s,·)θ(t +s,·)dx− W (t ,·)θ(t ,·)dx
G 0 0 G 0 0
Zx0−r Zx0−r
t0+s
+ f(·,x +r,W (·,x +r))θ(·,x +r)dt
0 G 0 0
(2.8)
Zt0
t0+s
− f(·,x −r,W (·,x −r))θ(·,x −r)dt
0 G 0 0
Zt0
+O(1)(s2+r2)(s+r+|u −u |)||θ|| ,
R L C1
where ∆(s,r;θ) is given in (2.6) and ||θ|| =||θ|| +||∂ θ|| +||∂ θ|| .
C1 C0 t C0 x C0
The left-hand side of (2.8) vanishes when W (t,x) is replaced by the exact solution of
G
R (u ,u ;t ,x ). Thus, the right hand side of (2.8) represents the error due to the choice of
G L R 0 0
approximate solution W (t,x).
G
Remark 2.2. 1. Condition (2.7) ensures that the waves in R (u ,u ;t ,x ) can not reach
C L R 0 0
the lines x = x ±r for t ≤ t +s, so that the waves in the rectangle region D ≡
0 0 (to,x0)
[x −r,x +r]×[t ,t +s) do not interact with the waves outside D .
0 0(cid:8) 0 0(cid:9) (to,x0)
2. In a different context, Liu [20] derived earlier an estimate similar to (2.7), but for an
approximation based on steady state solutions of the hyperbolic system and with initial data
consisting of two steady state solutions of (2.1) (with f =f(u) and g =g(x,u)).
3. Our formula (2.11) yields a possible generalization to the class of quasilinear systems
(1.1) of the notion of (classical) Riemann solver introduced by Harten and Lax in [14].
4. OnecanchecksimilarlythatW satisfiesanentropyinequalityassociatedwithanentropy
G
pair (when available). The error terms are completely similar to those found in (2.11). This
will be used to show that the weak solution generated by the random choice method satisfies
all the entropy inequalities.
Proof. Without loss of generality, we can assume that (t ,x )=(0,0). Given a C1 function
0 0
θ withcompactsupportinR ×R,wedefinem(t,x):=W ∂ θ+f(t,x,W )∂ θ+g(t,x,W )θ.
+ G t G x G
s r
From (2.6) we have ∆(s,r;θ)= m(t,x)dxdt. Next, we decompose ∆(s,r;θ) as
0 −r
R Rp
∆(s,r;θ)= ∆1(s,r;θ)+ ∆2(s,r;θ) (2.9)
i i
i=0 i−rare.
X wXaves
where
s σ− t
i+1
∆1(s,r;θ):= m(t,x)dxdt, 1≤i≤p−1,
i
Z0 Zσi+t
s σ−t s r
1
∆1(s,r;θ):= m(t,x)dxdt, ∆1(s,r;θ):= m(t,x)dxdt,
0 p
Z0 Z−r Z0 Zσp+t
and (if the i-wave, 1≤i≤p, is a rarefaction wave)
s σ+t
i
∆2(s,r;θ):= m(t,x)dxdt
i
Z0 Zσi−t
7
We firstcompute ∆1 inthe regionwhereclassicalRiemannsolutionW isaconstantstate.
i C
According to the form of W (t,x) in (2.5), it follows that
G
W (t,x)=u +tq(0,0;u ) (2.10)
G i i
for x ∈[σ+,σ− ],i∈ 1,2,...,p−1 . By asimple calculationandthe definitionofq in(1.4),
t i i+1
we have
(cid:8) (cid:9)
∂ W +∂ f(t,x,W )−g(t,x,W )=q(0,0;u )−q(t,x,W )
t G x G G i G
for i ∈ 1,2,...,p−1 . By multiplication by the function θ and then using integration by
parts, we obtain
(cid:8) (cid:9)
σ− s
i+1
∆1(s,r;θ) = W (s,x)θ(s,x)dx
i G
Zσi+s
s
+ (f(t,σ− t,W (t,σ− t))−σ− W (t,σ− t))θ(t,σ− t)dt
i+1 G i+1 i+1 G i+1 i+1
Z0 (2.11)
s
− (f(t,σ+t,W (t,σ+t))−σ+W (t,σ+t))θ(t,σ+t)dt
i G i i G i i
Z0
s σ− t
i+1
− q(0,0;u )−q(t,x,W ) θ(t,x)dxdt.
i G
Z0 Zσi+t
(cid:0) (cid:1)
By the property that q is Lipschitz continuous with respect to t, x and u on the compact set
[0,s]×[−r,r] and the formof W (t,x) in (2.10), the last term on the righthand side of (2.11)
G
can be estimated by O(s3)||θ|| with the bound O(1) depending on q. Therefore, equality
C0
(2.11) leads to
σ− s
i+1
∆1(s,r;θ) = W (s,x)θ(s,x)dx
i G
Zσi+s
s
+ (f(t,σ− t,W (t,σ− t))−σ− W (t,σ− t−))θ(t,σ− t)dt
i+1 G i+1 i+1 G i+1 i+1 (2.12)
Z0
s
− (f(t,σ+t,W (t,σ+t+0))−σ+W (t,σ+t+))θ(t,σ+t)dt
i G i i G i i
Z0
+O(1)s3||θ||
C0
for i=1,2,...,p−1. In the same fashion one can show that
σ−s 0
1
∆1(s,r;θ) = W (s,x)θ(s,x)dx− W (0,x)θ(0,x)dx
0 G G
Z−r Z−r
s
+ (f(t,σ−t,W (t,σ−t−))−σ−W (t,σ−t−))θ(t,σ−t)dt
1 G 1 1 G 1 1 (2.13)
Z0
s
− f(t,−r,W (t,−r))θ(t,−r)dt
G
Z0
+O(1)s2(s+r)||θ|| +O(1)sr2||θ|| ,
C0 C0
and
r r
∆1(s,r;θ) = W (s,x)θ(s,x)dx− W (0,x)θ(0,x)dx
p G G
Zσp+s Z0
s
+ f(t,r,W (t,r))θ(t,r)dt
G
(2.14)
Z0
s
− (f(t,σ+t,W (t,σ+t+))−σ+W (t,σ+t+))θ(t,σ+t)dt
p G p p G p p
Z0
+O(1)s2(s+r)||θ|| +O(1)sr2||θ|| .
C0 C0
8
Next, suppose that W (t,x) consists of an i-rarefaction wave in the region (t,x)|x ∈
C t
[σ−,σ+] for some i∈1,...,p. It follows that W (t,x) in this region is of the form
i i G (cid:8)
x x
(cid:9) W (t,x)=W ( )+tq(0,0;W ( ))
G C C
t t
whereWC(xt)isthei-rarefactionwaveofftheclassicalRiemfannproblemRC(uL,uR;t0,x0). By
setting ξ = x, W (t,x)=W (t,ξ), and the technique of change of variables (t,x)→(t,ξ), we
t G G
obtainf
∂ W +∂ f(t,x,W )f−g(t,x,W )
t G x G G
ξ 1
=∂ W − ∂ W + ∂ f(t,tξ,W )−g(t,tξ,W )
t G t ξ G t ξ G G (2.15)
1 ∂f ∂q dW
f f f f C
= (t,tξ,W )−ξI)(I +t (0,0,W ) · +q(0,0,W )−q(t,tξ,W )
G C C C
t ∂u ∂u dξ
(cid:16) (cid:17) f
where I is the p×pfidentity matrix. Since Wf (ξ) is a rarefactionfwave for thefsystem (2.3),
C
this implies that
f
1 ∂f dW
C
−ξ·I + (0,0,W ) · =0. (2.16)
C
t ∂u dξ
(cid:16) (cid:17) f
Thus, by applying (2.16) to (2.15) we obtain f
∂ W +∂ f(t,x,W )−g(t,x,W )
t G x G G
1 ∂f ∂f dW
C
= (t,x,W )− (0,0,W ) ·
t ∂u G ∂u C dξ (2.17)
(cid:16) (cid:17) f
∂f ∂q f dWC
+( (t,x,W )−ξI) (0,0,W )· +q(0,0,W )−q(t,x,W ).
G C C G
∂u ∂u dξ
f
Next, we multiply (2.17) by θ(t,x) and inftegrate the equationfover the region of i-rarefaction
wave: t < s and x ∈ [σ−,σ+]. Due to the Lipschitz continuity of ∂f and the fact that ∂f,
t i i ∂u ∂u
f
∂q, dWC and q remain bounded in [0,s]×[−r,r], the right hand side of (2.17) is bounded by
∂u dξ
O(1)s2(s+|u −u |). Therefore, by (2.17) again, we deduce the estimate
i i−1
σ+s
i
∆2(s,r;θ)= W (s,x)θ(s,x)dx
i G
Zσi−s
s
+ (f(t,σ+t,W (t,σ+t))−σ+W (t,σ+t))θ(t,σ+t)dt
i G i i G i i (2.18)
Z0
s
− (f(t,σ−t,W (t,σ−t))−σ−W (t,σ−t))θ(t,σ−t)dt
i G i i G i i
Z0
+O(1)s2(s+|u −u |)kθk .
R L C0
Next, note that an i-shock wave satisfies the Rankine-Hugoniot condition
f(0,0,u )−σ u =f(0,0,u )−σ u ,
i i i i−1 i i−1
and this implies that the approximate solution W (t,x) satisfies
G
s
[(f(t,σ t,W (t,σ t+))−σ W (t,σ t+))]θ(t,σ t)dt
i G i i G i i
Z0
s
(2.19)
− [(f(t,σ t,W (t,σ t−))−σ W (t,σ t−))]θ(t,σ t)dt
i−1 G i−1 i−1 G i−1 i−1
Z0
=O(1)s2|u −u |kθk
R L C0
9
where the bound O(1) depends on the Lipschitz constant of f and L∞-norm of q. Finally, by
the estimates (2.9), (2.12)-(2.14) and (2.18)-(2.19), we obtain
p
∆(s,t;θ)= ∆1(s,t;θ)+ ∆2(s,t;θ)
i i
i=0 i−rare.
X wXaves
σ−s p−1 σ− s
i i+1
= W (s,x)θ(s,x)dx+ W (s,x)θ(s,x)dx
G G
Z−r Xi=1Zσi+s
σ+s r
i
+ W (s,x)θ(s,x)dx+ W (s,x)θ(s,x)dx
G G
iw−Xarvaerse. Zσi−s Zσp+s
0 r
− W (0,x)θ(0,x)dx− W (0,x)θ(0,x)dx
G G
Z−r Z0
s s
+ f(t,r,W (t,r))θ(t,r)dt− f(t,−r,W (t,−r))θ(t,−r)dt
G G
Z0 Z0
+O(1)(s2+r2)(s+r+|u −u |) ||θ|| ,
R L C1
which leads to (2.8) and completes the proof.
3. Wave interaction estimates
In this section we study the nonlinear interaction of waves issuing from two Riemann solu-
tions and we derive estimates on the wave strengths.
WeemphasizethatthegeneralizedRiemannsolution,northeapproximatesolutionW (t,x)
G
of the generalized Riemann problem R (u ,u ;t ,x ) is not self-similar. The solution does
G L R 0 0
not consist of regions of constant value separated by straight lines. We thus should be careful
in defining the wave strengths In fact, we still define here the wave strengths by using the
underlying,classicalRiemannsolutionW (t,x). Wewillseelaterthatthisstrategyisaccurate
C
enoughandthatthediscrepancyintotalvariationbetweenW (t,x)andW (t,x)oneachtime
G C
stepisuniformly small(Cf.Section4)whenourGlimmschemeisappliedtothe problem(1.1),
(1.2). The sameobservationapplies to the potentialofwaveinteractionto be introducedlater.
In the rest of the section, all waves are considered as waves from some classical Riemann
problem unless specified otherwise. We say that an i-wave and a j-wave approach each other
(or interact in the future) if either i > j, or else i = j and at least one of two waves is a
shock wave. Suppose there are two solutions from different classical Riemann problems with
strengthsdenotedbyα=(α ,...,α )andβ =(β ,...,β ),thenthewaveinteractionpotential
i p i p
associated these two solutions is defined by
D(α,β):= |α β |, (3.1)
i j
(Xi,j)
where the notation (i,j) under the summation sign indicates an i-wave in one solution ap-
proaching a j-wave in the other solution, and the summation is on all approaching waves; also
α or β is negative when i = j. In addition, given a (u ,u ;t ,x ) ∈ U ×U ×R ×R, the
i i L R 0 0 +
wave strengths in R (u ,u ;t ,x ) are denoted by ε(u ,u ;t ,x ).
C L R 0 0 L R 0 0
We first recall:
10