RAIRO M ODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE
H ERVÉ G ILQUIN J ÉRÔME L AURENS C AROLE ROSIER Multi-dimensional Riemann problems for linear hyperbolic systems RAIRO – Modélisation mathématique et analyse numérique, tome 30, no 5 (1996), p. 527-548.
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MATHEMATICAL MODELUNG AND NUHERICAL AKALYSIS MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE
(Vol. 30, n° 5, 1996, p. 527 à 548)
MULTI-DIMENSIONAL RIEMANN PROBLEMS FOR LINEAR HYPERBOLIC SYSTEMS (*)
by Hervé GiLQUiN (*), Jérôme
LAURENS
(2) & Carole ROSIER (3)
Abstract. — In this paper we prove that each homogeneous linearfirst order hyperbolic system oftwo unknowns in N space dimensions with constant coefficients can be reduced to one ofthree canonical Systems. Then we give the explicit solution of the multi-dimensional Riemann problem associated with the most interesting canonical system on a structured or unstructured mesh. Résumé. — Nous montrons d'abord que tout système hyperbolique linéaire 2 x 2 à coefficients constants en N dimensions d'espace peut être réduit à un parmi trois systèmes canoniques. Nous donnons ensuite la solution explicite du problème de Riemann multi-dimensionnel associé au système canonique le plus intéressant, pour un maillage structuré ou non structuré.
1. INTRODUCTION
Let ( ^ O i ^ / ^ / y b e 2 x 2 matrices with constant coefficients; we are interested in solving the Riemann Problem for a linear system of partial differential équations
'='
xt t) = (
v
j (JC,
R2,
Vx = ( * ! , . . . , * „ ) e R " ,
A^.elR 2 * 2 ,
(1.1) l ^ i ^ N
W(x,r = 0 ) = W 0 (x)
(1.2)
(1.3)
(*) Manuscript received May 30, 1994 ; revised March 11, 1996. C) CNRS-ENS Lyon, Unité de Mathématiques Pures et Appliquées 46, Allée d'Italie F-69364 Lyon Cedex 07. (
[email protected]). (2) Département de Mathématiques, Laboratoire d'Analyse Appliquée et Optimisation, Université de Bourgogne, BP 138, F-21004 Dijon Cedex, (
[email protected]). (3) Laboratoire d'Analyse Numérique, Bâtiment 101, Université Claude Bernard Lyon I, F-69622 Villeurbanne Cedex, (
[email protected]) M 2 AN Modélisation mathématique et Analyse numérique 0764-583X/96/05/$ 4.00 Mathematical Modelling and Numerical Analysis © AFCET Gauthier-Villars
528
Hervé GILQUIN, Jérôme LAURENS, Carole ROSIER
where W0(x) is a simple piecewise constant vector valued function. The system is supposée to be hyperbolic according to [1] in the time direction for any 7V-uple a = (av .„, aN) e SN (the unit sphère in (RN), N
which means that for ail a, the matrix 2 =1
/
• C : Linear one-to-one change of unknowns. Let Q be a N x N invertible matrix with constant coefficients. The new unknowns W := QW satisfy :
i= 1
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Hervé GILQUÏN, Jérôme LAURENS, Carole ROSIER
multiplying by Q on the left, one obtains :
=0
with
À. = ÖA(. £T* •
(2.6)
System (2.6) still remains hyperbolic in the time direction because two conjugate matrices are diagonalizable over R at the same time and
The proof of Theorem 2.1 is carried out in eight steps. Step 1. One performs a galilean translation (action of type A) in order to set the trace of all the coefficient matrices to 0.
=0
with
Tr(Af.)=0.
(2.7)
Step 2. Then, either all matrices A. are null and the réduction is achieved (we have reached (2.1)), or at least one of them, Ai9 is not null. As System (2.7) is strictly hyperbolic in the coordinate a = (0,..., a/o = 1, ..., 0 ) e SN9 Aio must have two real eigenvalues with opposit signs (because (Tr {Af ) = 0 ) . Re-ordering coordinates with an action of type B exchanges Xf and xv so one can suppose that At is diagonizable with real eigenvalues and Tr (A 1 ) = 0. Step 3. An action of type C moves A« into a diagonalized form : there exists
i (x
°\
a 2 x 2 invertible matrix Q such that QAQ =1 Q _ ^ I. Then an action of type B introduces a new space coordinate xx := xx IX which leads to an equivalent hyperbolic System with :
The aim of the next steps is to reduce the other matrices A2 to AN. M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis
MULTI-DIMENSIONAL RIEMANN PROBLEMS
531
Step 4. A new action of type B changes x{ into xt - y. xx for all 2 ^ i: ^ Nt the flrst coordinate x1 remaining unchanged, the new equivalent system satisfies l
0
AA
j
(
y
(2.9)
Step 5. The spatial coordinates are ordered (action de type B) to obtain l
0
/O
r(o
0\
o)-
lf k = l the réduction of (1.1) is achieved (we have reached (2.3)). Step 6. If h > 1, one performs the change of space coordinates (action of type B) defined by :
x. : 1
which is well defined since fAiPi > 0 as the system is hyperbolic in the time direction for a = (0, ...> ai = 1,..., 0) ; one gets
(2.10)
a system with coefficients such that (2.10) holds remains hyperbolic in the time direction, thus, for each 2 < i ^ k ^ N, the matrices ô2 A2 - öt A. need to be diagonalizable which cannot be realized unless S2 — öi or equivalently A2 = Ai as we can see : 0 ô\-ô) vol. 30, n° 5, 1996
532
Hervé GILQUIN, Jérôme LAURENS, Carole ROSIER
//ll 0 0\ A ; gi Step 7. The change of unknowns defined by 0 == 1 A ) gives /O
0
1\
/o o\
(2.11)
Step 8. A last change of coordinates moves x- — x2 for ail 2 < i ^ k and complètes the proof. 0\
/O
j ^
1
j
/O /
0\
{
J
(2.12)
3. ANALYTIC SOLUTION OF THE MULTI-DIMENSIONAL RIEMANN PROBLEM FOR THE CANONICAL LINEAR SYSTEM
We look for the solution of the multi-dimensional Riemann problem for the three canonical Systems pointed out by the previous theorem. For Systems (2.1) and (2.3), one can easily exhibit the exact solution of any Cauchy problem as linear eombinations of plane waves with various propagation speed, so we focus on the interesting case (3.1). V(JC, y9t) G R 2 x R +
dtW(x,y,t)+AdxW(x,y,t)
+ BdyW(xfy,t)
=O
The two-dimensional elementary Riemann data are : W0(x,y) - W°++ H(x) H(y) + W°_
+
H(-x)
+ W°. _H(-x)H(-y)
H(y) + + W°+_ H(x)H(-y)
(3.2)
where W ++ refer to four constant vectors of M and H( . ) refers to the Heaviside function. THEOREM 3.1: When IR 2 xR + is décomposée into the six domains (Dt)l ^ l ^ 6 which sections at time t > 0 are given on figure 1, the solution of the multi-dimensional Riemann Problem (3.1)+ (3.2) is constant on every (D e ) 9 ^ • ^ 6, regular in D (inside a light cône), its values are given in table L M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis
MULTI-DIMENSIONAL RIEMANN PROBLEMS
t>0
D2
Ds
D3
533
—t D* -t Figure 1. — Plane décomposition associated to the solution of the Riemann problem (3.1) + (3.3).
Table 1. — Explicit formulae for u and v on a structured mesh.
Domain
u(xyy,t) = xy
-cos
D1
_
t2 - y2 —cos
-1
2?r
2 ir
cos
xy
-1
—X
cos
-1
2?r
D3
u°-v°
v°-u°
D5
v°-u°
vol. 30, n° 5, 1996
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534
Hervé GILQUIN, Jérôme LAURENS, Carole ROSIER
According to [4] and [5], System (3.1) is well posed in ail Hsloc(U2), (s e IR) because it is a linear symmetrie System with constant coefficients. Given initial data Wo in Hs]oc(U2), the solution W(x,y9.) belongs to k 2 y) ~dtv(.x,y,t = v°ô,
= 0)=- dyu0(x,y) + dvQ(x,y)
H(y)-uH(x)ôl
( 36 )
.
Thus the Kirchhof formula (3.9) is used to get explicit formulae that will be proven to define the unique solution of (3.1)+ (3.2). They rely on two auxiliary functions Fw(x,y, t) and Gw( xy y, t ) ( w e {M, V}) defined in the sensé of distribution by (3.7) and (3.8) :
Gw(x,y, t) := J - ^ ^ Û \/{x,y,t)e
R2xR\
(
^
WoUy)
(3.8)
w(x,y,t) = Fw(x,y,t) + dtGw(x,y,t) . (3.9) M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis
MULTI-DIMENSIONAL RffiMANN PROBLEMS
535
We now evaluate Fw and Gw in the domains denoted (Dl)l^i^6 (fig. 1) suggested by elementary considérations of finite propagation speed. For each domain, the results are given for both u(x, y> t) and v(x, y, t) but w e only carry out the calculations for the first one. • Dl := {(*, yt t) e U2 x U +
such
x2 + y2 ^ t2}.
that
Inside Z)\ we have :
- v° f
dn 2n
2TT
J
and similarly for u(x, v, r), Explicit formulae in terms of elementary functions follow from a simple calculations for Fu and the use of Gauss-Bonnet theorem (3.11) after a change in space variables for Gu. We can express the solution in terms of 0 and ¥ defined below : :=cos
-,
-b
Va, b such that a + b2 ^ 1 ,
' u^ a,v ^ b •
dudv
(3.10)
ab
4-
(3.11) one gets finally
and the solutions are :
V(x,y,t)eD\
2*~\rt) ..o /
2n~\t't ^
o (3.12)
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Hervé GILQUIN, Jérôme LAURENS, Carole ROSIER
One can easily verify that u(x, y> t) and v(x, y, t) belong to ^ ^ ( D 1 ) and are classical smooth solutions of (3.1) inside this light cone. • D2 :={(x,y, t) G R 2 x U+ such that x & t and y ^ t}, Inside D 2 , both functions Fu and Fv are identically 0 and
The same computation holds for Gv(x, y, t) = v° t thus : and ü(jc,y, r) = Ü° . D3 := {x2 + y2 5= r2, je ^ 0, y ^ 0} u
(3.13)
{y ^ - f}.
Inside D 3 , all of the functions FH, Fo, Gu and Gy are identically 0 and ,
(3.14)
w(x,y, 0 = i?(jc,y, f) s 0 .
D4 := {x2 + y2 2* t2, 0 ^ x ^ t, - t ^ y ^ 0} u {x ^ t, \y\ Inside D4 we have :
-t;0
2n
L;
Similar calculations give Fv(x, y, t) = ^ p - and G„(x, y, f ) = y ( j + r), so 4
° - V°
„o
o
D , u(x, y, t) = y^-=— and M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis
MULTI-DIMENSIONAL RIEMANN PROBLEMS
D5 := {
t2, - t
x ^ 0,0 ^ y ^ t}
KJ {\x\
5
537
^ t,y ^ t}.
4
This Domain D is treated in the same way as D , one has :
-u
\/(x,y,t)e
D5,
y, f)
=*-
u(x,ytt) = O and v(x, y, t) = v° .
D 6 : = { U } / , O ^ 2 x r , x2 + yz&t29
(3.16)
O ^ x ^ t md O ^ y ^ t).
6
Inside D , we obtain :
0
|V;=x
/
/»
v/l
= f^\ (f For v9 we get Fv(x>y, t) = V Zu finally gives
and
^,y, r ) =
which
(3.17)
and
We have to verify that W= T(u,v), defined in IR2 x 1R+ with initial data (3.2), is a weak solution of (3.1). W(x, y, t) must satisfy :
(3.18) dy0) =
, 0),
2 Define g = (-/?,/?) x [0, ( ) [ , T)) which contains the support pp of Let Dl := D' n Q and let ƒ*"'' = D* n D7 be the boundary separating Df and /y. We consider 3 the set of all (ij) such that T4^ has codim 1 : for
(ij)
e $, n\ is the normal to T*tJ from D' to D*(n\ = - wj). Let
vol. 30, n° 5, 1996
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Hervé GILQUIN, Jérôme LAURENS, Carole ROSIER
T W [W^
be
the
of W^D> on the boundary dDl and be the jump of W across fj, We finally set
trace
— TJW^J-^W^J
F* := Dl c\ dQ when it is not empty and dénote by n . the exterior normal to
r. For each test function 0 we integrate by parts the left hand side of (3.18) separately on each D' : T
W(dt0 + Adx& + Bd&)dxdy dt =
Q
T
&(dtW +
AdxW+BdW)dxdydt
ffT - [j T
.'tld + 'n .
—^ —^ —^
2
where ( x, y, t ) is the canonical basis of R x R . One has V ii,,
|
T
