Semi-discrete entropy satisfying approximate Riemann solvers. The case of the Suliciu relaxation approximation F. Bouchut∗, T. Morales de Luna† April 7, 2009
Abstract In this work we establish conditions for an approximate simple Riemann solver to satisfy a semi-discrete entropy inequality. The semi-discrete approach is less restrictive than the fully-discrete case and allows to grant some other good properties for numerical schemes. First, conditions are established in an abstract framework for simple Riemann solvers to satisfy a semi-discrete entropy inequality and then the results are applied, as a particular case, to the Suliciu system. This will lead in particular to the definition of schemes for the isentropic gas dynamics and the full gas dynamics system that are stable and preserve the stationary shocks.
Key-words: Entropy satisfying schemes; Riemann solvers; Conservation laws; Suliciu solver;
1
Introduction
The aim of this paper is to establish conditions for a simple Riemann solver to satisfy a semi-discrete entropy inequality. Classically, a discrete entropy inequality allows to fix a criteria in order to define weak solutions and to analyze the stability of a numerical scheme for a conservative system. Discrete-entropy inequalities have largely been studied for different conservative systems, see for example [4], [8]. Here we use a semi-discrete approach, where the time variable t is kept continuous and only the space ∗ D´ ´ epartement de Math´ ematiques et Applications, Ecole Normale 45 rue d’Ulm,75230 Paris cedex 05, France. Email:
[email protected] † Departamento de An´ alisis Matem´ atico, Facultad de Ciencias, Campus de Teatinons s/n, 29071 M´ alaga, Spain. Email:
[email protected]
1
variable x is discretized. This allows to have some stability properties for the scheme, formally when △t → 0, and gives a less restrictive approach. This leaves more choices in order to ask other properties like the conservation of stationary shocks by the scheme. Some schemes have already been presented for the full gas dynamics system that preserve the stationary shocks in [11],[5]. Here we aim entropy satisfying schemes that also preserve stationary shocks. In section 2 the basic definitions and results are given in a theoretical framework. We study then the Suliciu relaxation system in section 3 as a practical application to the results given by the previous section. The Suliciu relaxation system is attached to the resolution of the isentropic gas dynamics but can also be adapted to handle full gas dynamics, which will be shown in section 4. Finally, in section 5 we show some numerical results for the scheme described in section 3 and we compare to the results given by the fully discrete approach.
2
Entropy satisfying simple Riemann solvers
We introduce here the basic definitions and results we are going to use in next sections. We shall only give the details of those results that are new and we refer the reader to [4] for further details. We focus here on the study of one-dimensional conservative systems. We consider a one-dimensional system of conservation laws ∂t U + ∂x (F (U )) = 0,
t > 0, x ∈ R,
(2.1)
where U (t, x) is a vector with p components and F is a function that takes values in Rp . We assume that the system is hyperbolic, that is, A(U ) = F ′ (U ) has only real eigenvalues and a full set of eigenvectors. As usual, we define a first-order finite volume conservative scheme for (2.1): Uin+1 − Uin +
△t (Fi+1/2 − Fi−1/2 ) = 0, △xi
(2.2)
n where Fi+1/2 = F (Uin , Ui+1 ). It is a known fact that system (2.1) can have many weak solutions and entropies allow to fix a criteria in order to select a unique solution to the system.
Definition 2.1 An entropy for the system of conservation laws (2.1) is a convex function η(U ) with real values such that there exists another real valued function G(U ), called the entropy flux, satisfying G′ (U ) = η ′ (U )F ′ (U ). 2
(2.3)
Definition 2.2 A weak solution U (t, x) of (2.1) is said to be entropy satisfying for the pair entropy-entropy flux (η, G) if ∂t (η(U )) + ∂x (G(U )) ≤ 0.
2.1
(2.4)
Entropy satisfying numerical schemes
Defining a numerical scheme that satisfies (2.4) in some way is an interesting but a difficult task. Indeed, one may observe shocks in numerical solutions that are not entropy satisfying. At the same time a companion notion of stability for the numerical scheme is deduced from the existence of an entropy. In order to define numerical schemes that satisfy the entropy inequality, we introduce the following definitions. Definition 2.3 We say that the scheme (2.2) satisfies a discrete entropy inequality associated to the convex entropy η for (2.1), if there exists a numerical entropy flux function G(Ul , Ur ) which is consistent with the exact entropy flux (in the sense that G(U, U ) = G(U )), such that, under some Courant-Friedrichs-Lewy (CFL) condition, the discrete values computed by (2.2) automatically satisfy η(Uin+1 ) − η(Uin ) +
△t (Gi+1/2 − Gi−1/2 ) ≤ 0, △xi
(2.5)
with n Gi+1/2 = G(Uin , Ui+1 ).
(2.6)
Lemma 2.4 If a numerical scheme satisfies a discrete entropy inequality (Definition 2.3), then the following inequalities hold, G(Ur ) + η ′ (Ur )(F (Ul , Ur ) − F (Ur )) ≤ G(Ul , Ur ), G(Ul , Ur ) ≤ G(Ul ) + η ′ (Ul )(F (Ul , Ur ) − F (Ul )).
(2.7) (2.8)
Definition 2.5 We say that the scheme (2.2) satisfies a semi-discrete entropy inequality associated to the convex entropy η for (2.1), if there exists a numerical entropy flux function G(Ul , Ur ) which is consistent with the exact entropy flux (in the sense that G(U, U ) = G(U )), such that G(Ur ) + η ′ (Ur )(F (Ul , Ur ) − F (Ur )) ≤ G(Ul , Ur ), G(Ul , Ur ) ≤ G(Ul ) + η ′ (Ul )(F (Ul , Ur ) − F (Ul )).
(2.9) (2.10)
Remark 2.1 Definition 2.5 is related to the natural definition of an entropy inequality for a semi-discrete scheme where the time variable t is kept continuous and only the space variable x is discretized. We refer to [4] for further details. 3
Remark 2.2 Definition 2.3 seems the natural way to adapt (2.4) for numerical schemes and one would like to define numerical schemes that satisfies this property. Nevertheless, this is not an easy task and it can be too restrictive. Sometimes it is interesting to satisfy just the weaker conditions (2.9) and (2.10) which will allow us to ask the scheme to verify some other good properties as we will see in section 3.
2.2
Riemann solvers
We focus here in numerical schemes based on the notion of Riemann solvers and in particular we shall consider simple Riemann solvers. Definition 2.6 An approximate Riemann solver for (2.1) is a vector function R(x/t, Ul , Ur ) that is an approximation of the solution to the Riemann problem with initial data Ul , Ur , in the sense that it must satisfy the consistency relation R(x/t, U, U ) = U, (2.11) and the conservative identity Fl (Ul , Ur ) = Fr (Ul , Ur ), where Fl (Ul , Ur ) = F (Ul ) −
Z
(2.12)
0
(R(v, Ul , Ur ) − Ul ) dv,
(2.13)
(R(v, Ul , Ur ) − Ur ) dv.
(2.14)
−∞
Fr (Ul , Ur ) = F (Ur ) +
Z
∞ 0
Given an approximate Riemann solver R, we can define a conservative numerical scheme with the numerical flux F = Fl = Fr . Definition 2.7 An approximate Riemann solver for (2.1) is said to preserve the entropy shocks of the system if whenever Ul , Ur can be connected by an entropy shock, then R(x/t, Ul , Ur ) is the exact solution to the Riemann problem with initial data Ul , Ur . Lemma 2.8 Let R be an approximate Riemann solver and define the left and right discrete entropy fluxes by Z 0 Gl (Ul , Ur ) = G(Ul ) − (η(R(v, Ul , Ur )) − η(Ul )) dv, (2.15) −∞ Z ∞ Gr (Ul , Ur ) = G(Ur ) + (η(R(v, Ul , Ur )) − η(Ur )) dv. (2.16) 0
4
Suppose that Gr (Ul , Ur ) − Gl (Ul , Ur ) ≤ 0.
(2.17)
Then, under a half CFL condition |σl (Ui , Ui+1 )|△t ≤ △xi /2, σr (Ui−1 , Ui )△t ≤ △xi /2., the scheme associated to the approximate Riemann solver R satisfies (2.5) for any function G such that Gr (Ul , Ur ) ≤ G(Ul , Ur ) ≤ Gl (Ul , Ur ). Lemma 2.9 Let R be an approximate Riemann solver. Define the semidiscrete left and right entropy fluxes by
Gsl (Ul , Ur ) = G(Ul ) − η ′ (Ul )
Z
0
(R(v, Ul , Ur ) − Ul ) dv,
(2.18)
(R(v, Ul , Ur ) − Ur ) dv,
(2.19)
−∞ ∞
Gsr (Ul , Ur ) = G(Ur ) + η ′ (Ur )
Z
0
which are linearizations of (2.15)-(2.16). The scheme associated to R is semi-discrete entropy satisfying with respect to the convex entropy η if, and only if, Gsr (Ul , Ur ) − Gsl (Ul , Ur ) ≤ 0.
(2.20)
Proof. Suppose that (2.20) holds. Then, writing (2.13)-(2.14) into (2.9)-(2.10) we recover Definition 2.5 for any numerical entropy flux function G(Ul , Ur ) such that Gsr (Ul , Ur ) ≤ G(Ul , Ur ) ≤ Gsl (Ul , Ur ). (2.21) Conversely, if the scheme associated to R is semi-discrete entropy satisfying, from (2.9)-(2.10) and using (2.13)-(2.14) we recover (2.20). Definition 2.10 We call simple solver any approximate Riemann solver consisting of a set of finitely many simple discontinuities. This means that there exists a finite number m ≥ 1 of speeds σ0 = −∞ < σ1 < . . . < σm < σm+1 = +∞,
(2.22)
and intermediate states U0 = Ul , U1 , . . . , Um−1 , Um = Ur
(2.23)
such that R(x/t, Ul , Ur ) = Ui
if σi < x/t < σi+1 . 5
(2.24)
Remark 2.3 Given a simple solver defined by (2.22)-(2.24), define k ∈ {0, . . . , m} such that σk ≤ 0 < σk+1 . Then inequality (2.20) reads G(Ur ) − G(Ul ) m−1 X
′
≤ −η (Ur ) σk+1 (Uk − Ur ) +
(σi+1 − σi )(Ui − Ur )
i=k+1
− η ′ (Ul ) −σk (Uk − Ul ) +
k−1 X
!
!
(2.25)
(σi+1 − σi )(Ui − Ul ) .
i=1
Proposition 2.11 Consider a simple solver R with speeds (2.22) and intermediate states (2.23). Let k ∈ {0, . . . , m} such that σk ≤ 0 < σk+1 . A sufficient condition for (2.20) to hold is that there exist quantities Ei , i = 0, . . . , m satisfying G(Ur ) − G(Ul ) =
m−1 X
σi+1 (Ei+1 − Ei ),
(2.26)
i=0
Em = η(Ur ),
E0 = η(Ul ),
Ei ≥ η ′ (Ul )(Ui − Ul ) + η(Ul ), for i = 1, . . . , k − 1, k, Ei ≥ η ′ (Ur )(Ui − Ur ) + η(Ur ), for i = k, k + 1, . . . , m − 1.
(2.27) (2.28)
Proof. We rewrite
G(Ur ) − G(Ul ) = σm Em − σ1 E0 −
m−1 X
(σi+1 − σi )Ei .
i=1
6
(2.29)
Now, using the inequalities (2.28) we get G(Ur ) − G(Ul ) ≤ −
k−1 X
(σi+1 − σi )η ′ (Ul )(Ui − Ul )
i=1
− + +
m−1 X
(σi+1 − σi )η ′ (Ur )(Ui − Ur )
i=k+1 σk η ′ (Ul )(Uk
σk −
k−1 X (σi+1 − σi ) η(Ul ) i=1
+
− Ul ) − σk+1 η ′ (Ur )(Uk − Ur ) !
−σk+1 −
m−1 X
(2.30)
!
(σi+1 − σi ) η(Ur ) + σm Em − σ1 E0
i=k+1
= −η ′ (Ur ) σk+1 (Uk − Ur ) +
m−1 X
(σi+1 − σi )(Ui − Ur )
i=k+1
− η ′ (Ul ) −σk (Uk − Ul ) +
k−1 X
!
!
(σi+1 − σi )(Ui − Ul ) .
i=1
Thus, (2.25) holds and the result follows. Remark 2.4 There is an analogous result for fully discrete entropy inequalities. If we replace (2.28) by Ei ≥ η(Ui ),
for i = 1, ..., m − 1,
(2.31)
then (2.17) holds. η being convex, this condition is stronger than (2.28) as expected.
3
Modified Suliciu relaxation system
In this section we focus in the isentropic gas dynamics system ( ∂t ρ + ∂x (ρu) = 0, ∂t (ρu) + ∂x (ρu2 + p(ρ)) = 0,
(3.1)
where ρ ≥ 0 is the density of the gas, u is the speed, and p(ρ) is the pressure. We shall assume p(ρ) a strictly increasing function of ρ. We denote U = (ρ, ρu)t and we define η(U ) = ρu2 /2 + ρe(ρ),
G(U ) = (ρu2 /2 + ρe(ρ) + p(ρ))u, 7
(3.2)
where e′ (ρ) = p(ρ) ρ2 . (G, η) is a pair entropy flux-entropy for system (3.1). Using the so-called Suliciu relaxation system, which is introduced later, one may define a simple Riemann solver for (3.1) which is discrete entropy satisfying and preserves the positivity of the density ρ. This Suliciu system is described in [9],[10],[6],[3],[2],[4] and we refer to [4] for a detailed study. Nevertheless a discrete entropy satisfying condition can be too restrictive when we seek for some other properties. For example, one would like the Riemann solver to solve exactly the shocks of the original system. The aim is to modify the simple Riemann based on Suliciu system so that it satisfies just a semi-discrete entropy inequality, preserves positivity of ρ, and solves exactly the shocks of the system. Consider a Riemann problem for (3.1) with initial data ( (ρl , ρl ul ), if x < 0, (ρ, ρu)(t = 0, x) = (3.3) (ρr , ρr ur ), if x > 0. We introduce the Suliciu system ∂t ρ + ∂x (ρu) = 0, ∂ (ρu) + ∂ (ρu2 + π) = 0, t x 2 ∂t (ρπ/c ) + ∂x (ρπu/c2 ) + ∂x u = 0, ∂t (ρc) + ∂x (ρcu) = 0,
and we consider the Riemann problem with initial data ( (ρl ρl ul , ρl πl /c2l , ρl cl ), if x < 0, (ρ, ρu, ρπ/c2 , ρc) = , 2 (ρr , ρr ur , ρr πr /cr , ρr cr ), if x > 0,
(3.4)
(3.5)
where πl = p(ρl ),
πr = p(ρr ),
(3.6)
and cα ≡ cα (Uα ) α = l, r are positive and will be defined later on. The solution of (3.1)-(3.3) can be in general a tedious task. This is not the case of the solution of (3.4)-(3.5), which can be easily calculated as all the eigenvalues of the system are linearly degenerated. We shall denote f = (ρ, ρu, ρπ/c2 , ρc) and define R(x/t, fl , fr ) the exact solution to the Riemann problem for (3.4)-(3.5). This solution consists in three wave speeds σ1 = ul − cl /ρl , σ2 = u∗l = u∗r , σ3 = ur + cr /ρr , (3.7) with two intermediate states that we shall index l∗ and r∗ . The states are
8
obtained from the relations c∗l = cl , c∗r = cr , u∗l = u∗r , πl∗ = πr∗ , (π + cu)∗l = (π + cu)l , (π − cu)∗r = (π − cu)r ,
(3.8)
(1/ρ + π/c2 )∗l = (1/ρ + π/c2 )l , (1/ρ + π/c2 )∗r = (1/ρ + π/c2 )r . The solution is easily found to be cl ul + cr ur + πl − πr cr πl + cl πr − cl cr (ur − ul ) , πl∗ = πr∗ = , cl + cr cl + cr 1 cr (ur − ul ) + πl − πr cl (ur − ul ) + πr − πl 1 1 1 + + = , = . ∗ ∗ ρl ρl cl (cl + cr ) ρr ρr cr (cl + cr ) (3.9) We define the approximate simple Riemann solver for the system (3.1) u∗l = u∗r =
Rcl ,cr (v, Ul , Ur ) = LR(v, M (Ul , cl ), M (Ur , cr )),
(3.10)
where L : R4 → R2 is the canonical projection and M (U, c) = (ρ, ρu, ρp(ρ)/c2 , ρc).
(3.11)
The good properties of this solver depend on the definition of cl , cr which are assumed to be positive and depend on Ul , Ur . Some conditions have already been established in [4] for cl and cr in order to define a discrete entropy satisfying scheme. We intend to ask weaker conditions so that Rcl ,cr satisfies a semi-discrete entropy inequality and it preserves the shocks of system (3.1). We can consider a new variable e and add the equation ∂t (ρu2 /2 + ρe) + ∂x ((ρu2 /2 + ρe + π)u) = 0
(3.12)
to system (3.4). The solution of the Riemann problem for this new system is of the same type as the previous one consisting in three wave speeds with two intermediate states, determined by the relations (3.7)-(3.8) together with (e − π/2c2 )∗l = (e − π/2c2 )l ,
(e − π/2c2 )∗r = (e − π/2c2 )r .
(3.13)
We shall note by index 0, 1, 2, 3 the states l, l∗ , r∗ , r respectively. We notice then, according to this new equation, that σi (Ei − Ei−1 ) = ξi − ξi−1 , for i = 1, 2, 3, where E = ρu2 /2 + ρe and ξ = (ρu2 /2 + ρe + π)u. 9
(3.14)
Recall that here e is an independent variable. Thus, from (3.2), we get G(Ur ) − G(Ul ) = ξ3 − ξ0 =
2 X
σi+1 (Ei+1 − Ei ),
(3.15)
i=0
and we may apply Proposition 2.11 for this decomposition. Thus, (2.28) is a sufficient condition for the scheme to verify a semi-discrete entropy inequality for the pair entropy-entropy flux (η, G). Remark 3.1 • In [4] it is shown that under the hypothesis El∗ ≥ η(Ul∗ ), Er∗ ≥ η(Ur∗ ), the approximate Riemann solver Rcl ,cr satisfies an entropy inequality by interface. As η is a convex function, this hypothesis implies (2.28) as expected. • We will replace conditions (2.28) by Ei ≥ η(U ) + η ′ (U )(Ui − U ),
(3.16)
where U stands for either Ul or Ur , and Ui stands for either Ul∗ or Ur∗ . While this condition is stronger than (2.28), it will not impose too many limitations and simplifies further calculations. • As η(U ) = ρu2 /2 + ρe(ρ), one gets η(U ) − η(Ui ) + η ′ (U )(U − Ui ) = ρu2 /2 + ρe(ρ) − ρi u2i /2 − ρi e(ρi )+ (−u2 /2 + (e + p/ρ) (ρ))(ρi − ρ) + u(ρi ui − ρu) (ui − u)2 1 1 = ρi − . + e(ρ) − e(ρi ) + p(ρ) − 2 ρ ρi Thus, if we suppose ρi ≥ 0, condition (3.16) is equivalent to (ui − u)2 1 1 e(ρ) − ei + p(ρ) − − ≤ 0, ρ ρi 2
(3.17)
(3.18)
where (ρ, u) stands for either (ρl , ul ), (ρr , ur ) and (ρi , ui , ei ) for either (ρ∗l , u∗l , e∗l ), (ρ∗r , u∗r , e∗r ). Theorem 3.1 Let c2l =
(pr − pl )
1 ρl
−
1 ρr
(p − pl )2 h r − − 2(el − er ) + (pr + pl ) ρ1l − 10
1 ρr
i , +
(3.19)
c2r =
(pr − pl )
Then, for
1 ρl
−
1 ρr
(p − pl )2 h r − 2(el − er ) + (pr + pl ) ρ1l −
1 ρr
p cl = max cl , ρl (ul − ur )+ , ρl (pr − pl )+ , e p e cr = max cr , ρr (ul − ur )+ , ρr (pl − pr )+ ,
i .
(3.20)
+
(3.21) (3.22)
the simple solver Recl ,ecr described by (3.10) with wave speeds (3.7) and intermediate states (3.9), is an approximate Riemann solver for the isentropic gas dynamics system (3.1) and has the properties: (i) it preserves the non negativity of ρ, (ii) it satisfies a semi-discrete entropy inequality, (iii) it preserves the entropy shocks for the isentropic gas dynamics system. Remark 3.2 We recall that two different states Ul , Ur can be connected by a shock for system (3.1) if, and only if, 1 1 (ur − ul )2 = (pr − pl ) . (3.23) − ρl ρr An entropy 1-shock for (3.1) satisfies ul ≥ ur ,
ρr ≥ ρ l .
(3.24)
ρr ≤ ρ l .
(3.25)
An entropy 2-shock for (3.1) satisfies ul ≥ ur ,
We shall give some lemmas in order to prove this theorem. Lemma 3.2 Under the assumptions (pr − pl )2 2 el − er + pr ρ1l −
1 ρr
≤ c2l ,
condition (3.18) is satisfied.
(pr − pl )2 2 er − el + pl ρ1r −
1 ρl
≤ c2r , (3.26)
Remark 3.3 Using Taylor’s theorem, for any ρ, ρ0 ≥ 0, there exists ρe between ρ, ρ0 such that 2 1 1 d2 e 1 1 1 e(ρ) = e(ρ0 ) − p(ρ0 ) + − − . (3.27) ρ) 2 (e ρ ρ0 2 ρ ρ0 d ρ1 11
Assume p strictly increasing function of ρ. As e l − e r + pr
1 1 − ρl ρr
≥ 0,
d2 e d(1/ρ)2
er − e l + pl
= ρ2 dp dρ , we get
1 1 − ρr ρl
≥ 0.
(3.28)
Proof. In the case (U, Ui ) = (Ul , Ul∗ ) or (U, Ui ) = (Ur , Ur∗ ), from (3.9)-(3.13) it follows that (u − ui )2 1 1 − − e(ρ) − ei + p(ρ) ρ ρi 2 2 2 2 p(ρ) 1 πi p(ρ) πi (3.29) = − 2 (p(ρ) − πi )2 − + p(ρ) − 2c2 2c2 c2 c2 2c 1 = − 2 (p(ρ) − πi )2 ≤ 0 c and (3.18) is automatically satisfied. Consider now the case (U, Ui ) = (Ur , Ul∗ ) (the case (U, Ui ) = (Ul , Ur∗ ) being analogous).
12
Using again the identities (3.9)-(3.13) we get 1 (u∗ − ur )2 1 e(ρr ) − e∗l + p(ρr ) − ∗ − l ρr ρl 2
p2 − (πl∗ )2 = er − el + l 2c2l 1 pr − pl − cr (ur − ul ) 1 − + + pr ρr ρl cl (cl + cr ) 2 2 cr pr − pl + cl (ur − ul ) − 2 cr (cl + cr ) 1 pr − pl − cr (ur − ul ) 1 = e r − el + pr + (pr − pl ) − ρr ρl cl (cl + cr ) 2 2 1 pr − pl + cl (ur − ul ) 1 pr − pl − cr (ur − ul ) (3.30) − − 2 cl + cr 2 cl + cr 1 1 − = e r − el + pr ρr ρl 2 2 (ur − ul )2 c + cr cr (pr − pl )2 (pr − pl )(ur − ul ) − + l + − (cl + cr )2 2 cl (c2l + c2r ) cl 2 2 2 1 pr − pl cl + cr 1 = e r − el + pr u − u + − − r l ρr ρl 2(cl + cr )2 cl 1 + 2 (pr − pl )2 2cl 1 1 1 ≤ e r − el + pr + 2 (pr − pl )2 . − ρr ρl 2cl
Now, from Remark 3.3 and assuming (pr − pl )2 2 el − er + pr ρ1l −
we get e(ρr ) −
e∗l
+ p(ρr )
1 1 − ∗ ρr ρl
1 ρr
≤ c2l ,
−
(u∗l − ur )2 ≤ 0. 2
(3.31)
(3.32)
Lemma 3.3 Consider two different states (ρl , ρl ul ) and (ρr , ρr ur ) such that they can be connected by an entropy 1-shock (resp. 2-shock) for system (3.1) with shock speed σ. Then the following conditions are equivalent.
13
(i) The Suliciu relaxation system preserves the 1-shock (resp. preserves the 2-shock), (ii) cl =
pr −pl ul −ur
(resp. cr =
pl −pr ul −ur ).
Proof. First, we remark that the approximate Riemann solver (3.10) preserves a shock if and only if one of the following cases is satisfied: (A) Ul∗ = Ur∗ = Ul and σ = ur +
cr ρr ,
(B) Ul∗ = Ur∗ = Ur and σ = ul −
cl ρl ,
(C) Ul∗ = Ul , Ur∗ = Ur and σ = u∗l = u∗r . From (3.9), we see that the third case is not possible. Indeed, this would imply cr (ur − ul ) = pr − pl = −cl (ur − ul ). The constants cl , cr being positive, this would mean ul = ur and thus Ul = Ur . Now, from Remark 3.2 we get
ul −
cl cl ρr u r − ρl u l ρr (ul − ur ) cl − σ = ul − − = − ρl ρl ρr − ρl ρr − ρl ρl 1 pr − pl = − cl , ρl u l − u r
(3.33)
cr ρr u r − ρl u l ρl (ul − ur ) cr cr − σ = ur + − = + ρr ρr ρr − ρl ρr − ρl ρr 1 pr − pl = + cr . ρr u l − u r
(3.34)
and
ur +
cl and cr being positive, we get from Remark 3.2 that case (A) corresponds to 1-shocks and case (B) corresponds to 2-shocks. It is now clear that (i) implies (ii). Conversely, suppose that cl = (pr − pl )/(ul − ur ) (the case cr = (pl − pr )/(ul − ur ) being analogous). From Remark 3.2, cl =
pr − pl ul − ur = 1 1 . ul − ur ρl − ρr
Now, using this last identity in (3.9), it is easy to check that
14
(3.35)
u∗l = u∗r = ur , ρ∗l
= ρr ,
πl∗ = πr∗ = pr ,
(3.36)
ρ∗r = ρr ,
(3.37)
and the result follows. Lemma 3.4 Suppose ρl , ρr > 0 and p a strictly increasing convex function of ρ. Consider the definitions given by (3.19)-(3.20). (i) Assume ρl 6= ρr . Then cl , cr are well defined in the sense that they are non negative and the denominator is non null. p (ii) lim cl = lim cr = ρ p′ (ρ), ρl ,ρr →ρ
ρl ,ρr →ρ
(iii) If ul , ur are such that the states (ρl , ρl ul ), (ρr , ρr ur ) can be connected −pl by an entropy 1-shock for system (3.1), then cl = uprl −u . r (iv) If ul , ur are such that the states (ρl , ρl ul ), (ρr , ρr ur ) can be connected −pr . by an entropy 2-shock for for system (3.1), then cr = upll −u r (v) cl , cr satisfy inequality (3.26). Proof. We shall only give the proofs for cl , the case for cr being similar. In order to prove (i), consider the two possible cases: (A) 2(el − er ) + (pr + pl ) ρ1l − ρ1r ≥ 0. In this case the denominator in (3.19) reduces to (pr − pl ) ρ1l − ρ1r . p being an strictly increasing function of ρ it is now clear that this last expression is positive (B) 2(el − er ) + (pr + pl ) ρ1l − ρ1r < 0. The denominator coincides with 2 el − er + pr ρ1l − ρ1r , and from Remark 3.3 it follows that it is non-negative and non-null as long as ρl 6= ρr . This proves (i). Now, consider the function f (ρ) = 2(e(ρ) − e(ρr )) + (p(ρ) + p(ρr )) 15
1 1 − ρ ρr
,
(3.38)
for any ρ ∈ (0, ∞). Using Taylor’s theorem we get 1 d2 p (e ρ) f (ρ) = 2 d (1/ρ)2
for some ρe between ρ and ρr .
1 1 − ρe ρr
1 1 − ρ ρr
2
,
(3.39)
Thus
1 1 − ρl ρr 2 2 1 1 d p 1 1 1 (e ρ) = − − , 2 d (1/ρ)2 ρe ρr ρl ρr
2(el − er ) + (pr + pl )
for some ρe between ρl and ρr . Thus,
c2l =
dp d(1/ρ)
dp − d(1/ρ)
1 ρr
1 ρr
2 + o ρ1l − ρ1r 2 , 2 − ρ1l + o ρ1l − ρ1r
−
1 ρl
2
(3.40)
(3.41)
and (ii) follows. Moreover, as the pressure p is a convex function of ρ, 2(el − er ) + (pr + 1 1 pl ) ρl − ρr has the same sign as ρr − ρl . Thus, from remark 3.2, (iii) follows. (v) is trivial from the definitions of cl , cr .
Proof of theorem 3.1. First we remark that ρ∗l ≥ 0 as long as cl e cr e pr − pl ≥ (ul − ur ) + , ρl cl + e e cr e cl + e cr
(3.42)
which is equivalent to cl e c2 e e cr − (ul − ur ) ≥ pr − pl − l . (3.43) ρl ρl p Thus, the condition e cl ≥ max(ρl (ul − ur )+ , ρl (pr − pl )+ ) implies the non negativity of ρ∗l . The analogous can be said for ρ∗r and this proves (i). 16
From lemma 3.4 we get that e cl , e cr satisfy the semi-discrete entropy inequality (3.18), and this proves (ii). Finally, given a 1-shock for system (3.1), we have
c2l − ρl (pr − pl ) =
(pr − pl )2 pr − pl 1 − ρl (pr − pl ) = ≥ 0, (3.44) (ul − ur )2 1/ρl − 1/ρr ρr
and ρl (ul − ur ) = ρl
s
(pr − pl )
1 1 − ρl ρr
0, and specific entropy s(ρ, e).
17
(4.2)
We can define a family of entropies H = ρφ(s),
(4.3)
G = ρφu,
(4.4)
with entropy fluxes where φ is an arbitrary function such that H is convex with respect to the conservative variables (ρ, ρu, ρ(u2 /2 + e)). Remark 4.1 A necessary condition for H in (4.3) to be convex with respect to (ρ, ρu, ρ(u2 /2 + e)) is that φ′ ≤ 0. Conversely, if −s is a convex function of (1/ρ, e) and if φ′ ≤ 0 and φ′′ ≥ 0, then H is convex. The full gas dynamics system and the definition of an entropy satisfying scheme can be handled via a general idea introduced in [6] adapted here to the semi-discrete case.
4.1
Reduction to an almost isentropic system
The idea consists in reversing the role of energy conservation and entropy inequality. We shall consider the system ∂t ρ + ∂x (ρu) = 0, (4.5) ∂t (ρu) + ∂x (ρu2 + p) = 0, ∂t (ρs) + ∂x (ρsu) = 0,
with the entropy inequality
∂t (ρ(u2 /2 + e)) + ∂x ((ρ(u2 /2 + e) + p)u) ≤ 0.
(4.6)
The convexity of this entropy with respect to (ρ, ρu, ρs) is ensured by the physically relevant condition −s is a convex function of (1/ρ, e).
(4.7)
Now suppose that we have a conservative numerical scheme for solving (4.5), △t ρ ρ (F − Fi−1/2 ) = 0, ρn+1 − ρi + i △x i+1/2 △t ρu ρu ρn+1 un+1 − ρi u i + (Fi+1/2 − Fi−1/2 ) = 0, (4.8) i i △x △t ρs ρs ρn+1 sn+1 − ρi si + (F − Fi−1/2 ) = 0, i i △x i+1/2 18
satisfying a semi-discrete entropy inequality for η(U, ρs) = ρu2 /2 + ρe(ρ, s),
G(U, ρs) = (ρu2 /2 + ρe(ρ, s) + p(ρ, s))u, (4.9)
which means G(Ur , ρr sr ) − G(Ul , ρl sl ) + ∇(U,ρs) η(Ur , ρr sr )(F (ρ,ρu,ρs) (Ul , ρl sl , Ur , ρr sr ) − F (ρ,ρu,ρs) (Ur , ρr sr )) − ∇(U,ρs) η(Ul , ρl sl )(F (ρ,ρu,ρs) (Ul , ρl sl , Ur , ρr sr ) − F (ρ,ρu,ρs) (Ul , ρl sl )) ≤ 0. (4.10) We assume moreover that the scheme satisfies for any φ convex the inequalities G(Ur , ρr sr ) − G(Ul , ρl sl ) + ∇(U,ρs) H(Ur , ρr sr )(F (ρ,ρu,ρs) (Ul , ρl sl , Ur , ρr sr ) − F (ρ,ρu,ρs) (Ur , ρr sr )) − ∇(U,ρs) H(Ul , ρl sl )(F (ρ,ρu,ρs) (Ul , ρl sl , Ur , ρr sr ) − F (ρ,ρu,ρs) (Ul , ρl sl )) ≤ 0. (4.11) Define now the scheme for the gas dynamics system (4.1) △t ρ ρ ρn+1 − ρi + (Fi+1/2 − Fi−1/2 ) = 0, i △x △t ρu ρu ρn+1 un+1 − ρi u i + (F − Fi−1/2 ) = 0, i i △x i+1/2 E E Ein+1 − Ei + △t (Fi+1/2 ) = 0, − Fi−1/2 △x
(4.12)
2
where E = ρ u2 + ρe, and F E is any flux verifying G(Ur , ρr sr )
+ ∇(U,ρs) η(Ur , ρr sr )(F (ρ,ρu,ρs) (Ul , ρl sl , Ur , ρr sr ) − F (ρ,ρu,ρs) (Ur , ρr sr )) ≤ F E (Ul , ρl sl , Ur , ρr , sr ) ≤ G(Ul , ρl sl ) + ∇(U,ρs) η(Ul , ρl sl )(F (ρ,ρu,ρs) (Ul , ρl sl , Ur , ρr sr ) − F (ρ,ρu,ρs) (Ul , ρl sl )). (4.13) Of course, we take initially si = s(ρi , ei ).
Proposition 4.1 The scheme (4.12) verifies a semi-discrete entropy inequality for the entropy pair H(U, E) = ρφ(s(ρ, e)),
G(U, E) = ρφ(s(ρ, e))u, 19
(4.14)
i.e., G(Ur , Er ) − G(Ul , El ) + ∇(U,E) H(Ur , E)(F (ρ,ρu,E) (Ul , El , Ur , Er ) − F (ρ,ρu,E) (Ur , Er )) − ∇(U,E) H(Ul , El )(F (ρ,ρu,E) (Ul , ρl sl , Ur , Er ) − F (ρ,ρu,E) (Ul , El )) ≤ 0. (4.15) Proof. Let S(U, E) = ρs(ρ, e). Using the relations H(U, E) = H(U, ρs(ρ, e)), G(U, E) = G(U, ρs(ρ, e)), ∇(U,E) H(U, E) = (∇U H(U, ρs(ρe)), 0) ∂H + (U, ρs(ρ, e))∇(U,E) S(U, E), ∂(ρs) −1 ∂η · (∇U η(U, ρs), −1) , ∇(U,E) S(U, η(U, ρs)) = − ∂(ρs)
(4.16)
we decompose the left hand side of (4.15) in three sum terms (I), (II), (III), (I) = G(Ur , ρr sr ) − G(Ul , ρl sl ) + ∇(U,ρs) H(Ur , ρr sr ) F (ρ,ρu,ρs) (Ul , ρl sl , Ur , ρr sr )
− F (ρ,ρu,ρs) (Ur , ρr sr )
− ∇(U,ρs) H(Ul , ρl sl ) F (ρ,ρu,ρs) (Ul , ρl sl , Ur , ρr sr )
(4.17)
− F (ρ,ρu,ρs) (Ul , ρl sl ) ,
(II) =
−
∂H (Ur , ρr sr )∇(U,E) S(Ur , Er ) F (ρ,ρu,E) (Ul , ρl sl , Ur , ρr sr ) ∂(ρs) − F (ρ,ρu,E) (Ur , ρr sr )
∂H (Ur , ρr sr )(F ρs (Ul , ρl sl , Ur , ρr sr ) − F ρs (Ur , ρr sr )), ∂(ρs)
(4.18) ∂H (Ul , ρl sl )∇(U,E) S(Ul , El ) F (ρ,ρu,E) (Ul , ρl sl , Ur , ρr sr ) (III) = − ∂(ρs) − F (ρ,ρu,E) (Ul , ρl sl ) +
∂H (Ul , ρl sl )(F ρs (Ul , ρl sl , Ur , ρr sr ) − F ρs (Ul , ρr sl )). ∂(ρs)
(4.19) 20
We show that each of them is non-positive. From (4.11), it follows that (I) ≤ 0. We shall prove (II) ≤ 0, the inequality (III) ≤ 0 being similar. Using (4.13) and (4.16) one gets (II) = −φ′ (sr )
∂s (ρr , er )(∇U η(Ur , ρr sr ), −1) F (ρ,ρu,E) (Ul , ρl sl , Ur , ρr sr ) ∂e
− F (ρ,ρu,E) (Ur , ρr sr ) − φ′ (sr )(F ρs (Ul , ρl sl , Ur , ρr sr ) − F ρs (Ur , ρr sr )) ∂s = −φ′ (sr ) (ρr , er ) −F E (Ul , ρl sl , Ur , ρr sr ) + F E (Ur , ρr sr ) ∂e + ∇(U,ρs) η(Ur , ρr sr ) F (ρ,ρu,ρs) (Ul , ρl sl , Ur , ρr sr ) − F (ρ,ρu,ρs) (Ur , ρr sr ) ≤ 0.
(4.20)
And this completes the proof.
4.2
Resolution of an extended Suliciu relaxation system
In order to solve system (4.5) we propose an extension of system (3.4), (3.12). We add the transport of specific entropy, ∂t ρ + ∂x (ρu) = 0, ∂t (ρu) + ∂x (ρu2 + π) = 0, ∂ (ρu2 /2 + ρe) + ∂ ((ρu2 /2 + ρe + π)u) = 0, t x (4.21) 2 ∂t (ρπ/c ) + ∂x (ρπu/c2 ) + ∂x u = 0, ∂t (ρc) + ∂x (ρcu) = 0, ∂t (ρs) + ∂x (ρsu) = 0.
One has to take care that in (4.21), ρ, e, s, π are understood as independent variables. The solution to the Riemann problem for (4.21) is obvious since s is not coupled, and it has the same structure as before: (3.9), (3.13), to which we prescribe s∗l = sl , s∗r = sr . Only the initialization is modified, now πl = pl = p(ρl , sl ), πr = pr = p(ρr , sr ). (4.22) We shall define as before the simple Riemann solver Rcl ,cr (v, Ul , ρl sl , Ur , ρr sr ) as the canonical projection on the variables (U, ρs) of the exact solution of the corresponding Riemann problem for (4.21) with (4.22) and cl , cr to be defined later on.
21
Using the same approach as in the isentropic case, we consider the decomposition
G(Ur , ρr sr ) − G(Ul , ρl sl ) = ξ3 − ξ0 =
3 X
σi (Ei − Ei−1 ),
(4.23)
i=1
with E = ρu2 /2 + ρe and ξ = (ρu2 /2 + ρe + π)u. Recall that here e is an independent variable. Then, from Proposition 2.11, a sufficient condition for (4.10) to hold is Ei ≥ ∇(U,ρs) η(U, ρs)
Ui −U ρi si −ρs
+ η(U, ρs),
(4.24)
where (U, ρs) stands for either (Ul , ρl sl ) or (Ur , ρr sr ), and the i-indexed quantities stand for either (Ul∗ , ρ∗l sl ) or (Ur∗ , ρ∗r sr ). Using the expression of η and assuming ρ∗l , ρ∗r ≥ 0, condition (4.24) can be replaced by 1 (ui − u)2 1 ∂e e(ρ, s) − ei + p(ρ, s) − − + (ρ, s)(si − s) ≤ 0, (4.25) ρ ρi 2 ∂s with (ρ, u, s) ∈ {(ρl , ul , sl ), (ρr , ur , sr )}, (ρi , ui , ei , si ) ∈ {(ρ∗l , u∗l , e∗l , s∗l ), (ρ∗r , u∗r , e∗r , s∗r )}.
(4.26)
We shall note τ = 1/ρ and we rewrite relation (4.2) under the form de = −pdτ + T ds,
(4.27)
being e the internal energy, p the pressure and T the temperature. ∂p Definition 4.2 Define ∂τ by the relation r
p +p ∂p pr − pl p(τr , sl ) − p(τr , sr ) el − er − l 2 r (τr − τl ) =− (4.28) · − τl − τr ∂τ r e(τr , sl ) − e(τr , sr ) τl − τr ∂p We may also define ∂τ by exchanging the labels l, r in the previous defil nition. Remark 4.2 p(τr , sl ) − p(τr , sr ) ∂p lim = sr ,sl →s e(τr , sl ) − e(τr , sr ) ∂s
22
∂e ∂s
−1
(τr , s) =
∂p (τr , e(τr , s)), ∂e (4.29)
Thus, p(τr , sl ) − p(τr , sr ) (el −er ) = (τl −τr )ϕ, with ϕ a regular function, e(τr , sl ) − e(τr , sr ) (4.30) ∂p and ∂τ is well defined.
pl −pr −
r
Theorem 4.3 Let c2l
= max
∂p , − ∂τ l
! (pr − pl )2 , (ρ , s )(s − s )) − ρ1r + ∂e r r r l ∂s
(4.31)
! (pr − pl )2 , ∂e (ρl , sl )(sl − sr )) 2(er − el + pl ρ1r − ρ1l + ∂s
(4.32)
2(el − er + pr c2r
= max
Then, for
∂p − , ∂τ r
cl = max cl , ρl (ul − ur )+ , e (ur − ul )+ +
1 ρl
p ρl (pr − pl )+ ),
q ! ((ur − ul )+ )2 + 4 pell (pl − pr )+ 2el /pl
e cr = max cr , ρr (ul − ur )+ , (ur − ul )+ +
p ρr (pl − pr )+ ,
(4.33) ,
q ! ((ur − ul )+ )2 + 4 perr (pr − pl )+ , 2er /pr
(4.34)
the simple solver Recl ,ecr defined by the wave speeds (3.7) and intermediate states (3.9), (3.13), with (4.22) is an approximate Riemann solver for the full gas dynamics system (4.1) and satisfies the following properties: (i) it preserves the non negativity of ρ, (ii) it preserves the non negativity of e,
23
(iii) it satisfies a semi-discrete entropy inequality, (iv) it preserves the entropy 1 and 3 shocks for the full gas dynamics system (4.1). Remark 4.3 Let (ρl , ul , el ), (ρr , ur , er ), such that (ur −ul )(1/ρr −1/ρl ) 6= 0. We recall that these two states can be connected by a shock for the full gas dynamics system (4.1) if, and only if, the following two relations hold, (ur − ul )2 = (pr − pl ) ρ1l − ρ1r , (4.35) 1 1 l er − el = pr +p (4.36) 2 ρl − ρr . The entropy 1-shocks for (4.1) satisfy ul ≥ ur ,
ρr ≥ ρ l ,
pr ≥ pl ,
sr ≥ s l ,
er ≥ el .
(4.37)
sr ≤ s l ,
er ≤ el .
(4.38)
The entropy 3-shocks for (4.1) satisfy ul ≥ ur ,
ρr ≤ ρ l ,
pr ≤ pl ,
To prove this theorem, we shall prove some lemmas. Lemma 4.4 Under the assumptions c2l ≥ c2r
≥
2(el − er + pr ( ρ1l
(pr − pl )2 , − ρ1r ) + ∂e ∂s (ρr , sr )(sr − sl ))
2(er − el + pl ( ρ1r
(pr − pl )2 , − ρ1l ) + ∂e ∂s (ρl , sl )(sl − sr ))
(4.39)
condition (4.25) is satisfied. Proof. In the cases (U, ρs, Ui , ρi si ) = (Ul , ρl sl , Ul∗ , ρ∗l s∗l ) and (U, ρs, Ui , ρi si ) = (Ur , ρr sr , Ur∗ , ρ∗r s∗r ), we have s−si = 0 and following analogous calculations as in the isentropic case, one gets
e(ρ, s) − ei + p(ρ, s)
1 1 − ρ ρi
1 = − 2 (p(ρ, s) − πi )2 ≤ 0. c
−
(ui − u)2 ∂e + (si − s) 2 ∂s
(4.40)
We show now the case (U, ρs, Ui , ρi si ) = (Ur , ρr sr , Ul∗ , ρ∗l s∗l ) the other case being similar. Some simple calculations show 24
1 (u∗ − ur )2 1 ∂e ∗ e(ρr , sr ) − e∗l + p(ρr , sr ) − ∗ − l + (s − sr ) ρr ρl 2 ∂s l 1 1 1 = e r − e l + pr + 2 (pr − pl )2 − (4.41) ρr ρl 2cl 2 ∂e pr − pl c2l + c2r + u − u + (ρr , sr )(sl − sr ) − . r l ∂s 2(cl + cr )2 cl Using Taylor’s theorem, −τr e(τl , sl ) = e(τr , sr ) + (−p(τr , sr ), T (τr , sr )) sτll −s r τl −τr 2 2 ∗ ∗ + Dτ,s e(τ , s ) sl −sr ,
(4.42)
for some (τ ∗ , s∗ ) in the segment defined by (τl , sl ), (τr , sr ). As −s(τ, e) is a convex function, then e(τ, s) is convex and we get ∂e 1 1 + − (ρr , sr )(sl − sr ) ≤ 0, (4.43) er − e l + pr ρr ρl ∂s and the result follows. Lemma 4.5 Let (ρl , ρl ul , El ),(ρr , ρr ur , Er ) two different states such that they can be connected by an entropy 1-shock (resp. 2-shock) for the full gas dynamics system (4.1) with shock speed σ. Then the following conditions are equivalent. (i) The Suliciu relaxation system preserves the 1-shock (resp. preserves the 3-shock), (ii) cl =
pr −pl ul −ur
(resp. cr =
pl −pr ul −ur ).
Proof. The proof can be stated in the same way as in the isentropic case. Lemma 4.6 Let cl , cr be defined by (4.31)-(4.34). Then the following properties hold: s ∂p (i) lim cl = lim cr = − 1 (ρ, s) ∂( ρ ) ρ l , ρr → ρ ρl , ρr → ρ s r , sl → s s r , sl → s 25
(ii) If (ρl , ul , el ), (ρr , ur , er ) can be connected by an entropy 1-shock for −pl (4.1), then cl = uprl −u , r (iii) If (ρl , ul , el ), (ρr , ur , er ) can be connected by an entropy 3-shock for r (4.1), then cr = upll −p −ur , (iv) cl , cr satisfy the inequalities (4.39). Proof.
−
pl + pr p(ρr , sr ) − p(ρr , sl ) ∂p , = − ϕ − · ∂τ l 2 e(ρr , sr ) − e(ρr , sl )
(4.44)
r ,sr )−p(ρr ,sl ) + p(ρr , sl ) p(ρ e(ρr ,sr )−e(ρr ,sl ) + O(ρr − ρl ). ∂p ∂p Thus, we get the limit − ∂τ for ρr − ρl → 0 and sr − sl → 0. → − ∂τ
where ϕ =
∂p 1 (ρr , sl ) ∂( ρ )
l
The convexity of e(τ, s) implies ∇p(τ, s)
x 2 y
≤−
∂p (τ, s) · D2 e(τ, s) ∂τ
x 2 , y
for all
x y
∈ R2 .
(4.45)
Thus, there exist (τ1 , s1 ), (τ2 , s2 ) in the segment defined by (τl , sl ), (τr , sr ) such that, (pr − pl )2 2(el − er + pr ρ1l − ρ1r + ∂e ∂s (ρr , sr )(sr − sl )) 2 −τl ∇p(τ1 , s1 ) sτrr −s l = −τl 2 D2 e(τ2 , s2 ) sτrr −s l −τl 2 D2 e(τ1 , s1 ) sτrr −s ∂p ∂p l ≤ − (ρ1 , s1 ) →− , τr −τl 2 2 ∂τ ∂τ D e(τ2 , s2 )
0≤
(4.46)
sr −sl
and (i) follows. From lemma 4.5 and from remark 4.3, (ii) and (iii) are easily verified. Finally, (iv) is trivial. Proof of theorem 4.3. p As in the isentropic case, the condition e cl ≥ max(ρl (ul −ur )+ , ρl (pr − pl )) implies the non negativity of ρ∗l . The analogous can be said for ρ∗r .
26
We shall show e∗l ≥ 0 (e∗r ≥ 0 is proved in the same way). This is equivalent to 2 cl e (pr − pl − e cr (ur − ul ))2 2e c2l el ≥ − cl + e e cr (4.47) cl e − 2pl (pr − pl − e cr (ur − ul ). cl + e e cr Thus, a sufficient condition for e∗l ≥ 0 to hold is 2e c2l el ≥ −2pl and, as we have
cl e (pr − pl − e cr (ur − ul )), cl + e e cr
1 −pl (pr − pl − e cr (ur − ul )) ≤ pl cl + e e cr
(4.48)
(pl − pr )+ + (ur − ul )+ , (4.49) e cl
a sufficient condition for (4.48) to hold is (pl − pr )+ e cl e l ≥ p l + (ur − ul )+ e cl
which is satisfied if we take q (ur − ul )+ + ((ur − ul )+ )2 + 4 pell (pl − pr )+ , cl ≥ e 2 pell
(4.50)
(4.51)
and this proves (ii). (iii) is trivial from lemma 4.4. Finally, for an entropy 1-shock for the system (4.1), we have pr ≥ pl , ul ≥ ur , and for an entropy 3-shock we have pl ≥ pr , ul ≥ ur . Thus,(iv) can be established in the same way as in the isentropic case. Remark 4.4 As in the isentropic case, the non-negativity of the variables ρ, e has been established in a discrete sense. (See remark 3.4).
5
Numerical tests
We show here some tests for the scheme described in section 3. For all tests we take p(ρ) = gρ2 /2 with g = 9.81. We use 100 point in the specified domain. 27
Suliciu scheme Lax-Friedrichs Force GForce
1.0 0.9
t = 5.00
t = 5.00
2.10
Suliciu scheme Lax-Friedrichs Force GForce
2.05
0.8 2.00 0.7 1.95
0.6 0.5
0.4
0.2
0.0
0.2
1.90
0.4
0.4
0.2
(a) Density
0.0
0.2
0.4
(b) Discharge
Figure 1: Numerical simulations corresponding to (5.1). Comparison with Lax-Friedrichs, Force and GForce schemes First, we consider the following Riemann problem in the interval [−0.5, 0.5] with initial data q ( 2 g 3 , if x < 0, 0.5, if x < 0, 2 ρ0 (x) = u0 (x) = q (5.1) g 3, 1, if x > 0, if x > 0. 2
The left and right states can be connected by a stationary entropy shock. We know that the modified Suliciu scheme described in section 3 preserves such shocks as it is shown in Figure 1 and Figure 2, while it is not the case for other schemes shown here. We remark that Roe scheme would preserve also such shocks but its computational cost is more expensive than that of Suliciu scheme. Moreover, Roe scheme does not satisfy in general an entropy inequality; as a consequence, an entropy-fix technique has to be added to the numerical scheme in order to capture the entropy solution in the presence of smooth transitions while it is not the case for the Suliciu scheme. Figure 1 compares the scheme defined in section 3 with classical Lax-Friedrichs scheme and Force and Gforce schemes introduced in [12], while Figure 2 compares the Suliciu scheme to Lax-Friedrichs combined with a third order WENO reconstruction or the hyperbolic reconstruction introduced in [13] as well as the GForce scheme combined with an hyperbolic reconstruction. Let us consider now the a Riemann problem in the interval [0, 1] with initial data ( ( 1, if x < 0.5, 3.2125, if x < 0.5, 0 0 ρ (x) = u (x) = (5.2) 2, if x > 0.5, 0.5, if x > 0.5.
28
t = 5.00 Suliciu scheme LF+WENO3 LF+hyp. reconst. GForce+hyp. reconst.
1.0 0.9
t = 5.00
2.10
2.05
Suliciu scheme LF+WENO3 LF+hyp. reconst. GForce+hyp. reconst.
0.8 2.00 0.7 0.6
1.95
0.5
0.20
0.15
0.10
0.05
0.00
0.05
0.10
0.15
1.90 0.20
0.20
(a) Density
0.15
0.10
0.05
0.00
0.05
0.10
0.15
0.20
(b) Discharge
Figure 2: Numerical simulations corresponding to (5.1). Comparison with High Order schemes The exact solution to this Riemann consists of a 1-shock of speed σ = −2.2125. Numerical results are shown in Figure 3, Figure 4 and Figure 5. Figure 3 compares the modified Suliciu scheme introduced in section 3 with the original discrete entropy satisfying Suliciu scheme. Figure 4 and Figure 5 compare the scheme with other numerical fluxes. As we can see, the semidiscrete approach give less diffusive shock profiles than the fully discrete approach and the accuracy of the scheme in shocks is comparable to that of high order reconstruction schemes. Now, we solve the Riemann problem with initial data ( ( 1 if x < 0.5, 1 if x < 0.5, 0 0 ρ (x) = u (x) = (5.3) 1.2242 if x > 0.5, 1.666 if x > 0.5. . The solution consist of a simple rarefaction wave. As we can see in Figure 6, no major differences is shown in this case for the two Suliciu schemes as the improvement of the modified Suliciu scheme is related to shock regions. Figure 7 compares the scheme with different high order reconstruction techniques. As expected, a high order scheme will give better results in smooth regions. Finally we would like to show an example to enlighten that the modified Suliciu scheme can be useful also in the context of Saint Venant equations with arbitrary topography ∂t h + ∂x (hu) = 0, (5.4) ∂t (hu) + ∂x (hu2 + g2 h2 ) = −gh∂x z, where h(t, x) ≥ 0 is the water height, u(t, x) is the velocity and z(x) is a 29
2.0
t = 0.10 Exact Semi-disc. ent. satisf. Disc. entropy satisfying
3.0
1.8
2.5
1.6 1.4
2.0
1.2
1.5
1.0 0.20
t = 0.10
3.5
0.22
0.24
0.26
0.28
0.30
0.32
1.0 0.20
0.34
Exact Semi-disc. ent. satisf. Disc. entropy satisfying 0.22
(a) Density
0.24
0.26
0.28
0.30
0.32
0.34
(b) Discharge
Figure 3: Numerical simulations corresponding to (5.2). Comparison of semi-discrete entropy satisfying Suliciu scheme with the fully discrete entropy satisfying Suliciu scheme
t = 0.10
t = 0.10
3.5
Exact 2.0
Modified Suliciu Force
1.8
3.0
Gforce
2.5 1.6
2.0
1.4
1.2
1.5
Exact Modified Suliciu Force
1.0 1.0 0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.20
(a) Density
Gforce 0.22
0.24
0.26
0.28
0.30
0.32
0.34
(b) Discharge
Figure 4: Numerical simulations corresponding to (5.2). Comparison with Lax-Friedrichs, Force and GForce schemes
30
t = 0.10
t = 0.10
3.5
Exact 2.0
Modified Suliciu LF+WENO3
1.8
3.0
LF+hyp. reconst. GForce+hyp. reconst.
2.5 1.6
2.0
1.4
Exact 1.2
1.5
Modified Suliciu LF+WENO3 LF+hyp. reconst.
1.0 1.0 0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.20
GForce+hyp. reconst. 0.22
(a) Density
0.24
0.26
0.28
0.30
0.32
0.34
(b) Discharge
Figure 5: Numerical simulations corresponding to (5.2). Comparison with High Order schemes
1.25 1.20 1.15
t = 0.05 Exact Suliciu (semi-disc. ent. satisf.) Suliciu (disc. ent. satisf.) Force GForce
2.0 1.8 1.6
1.10
1.4
1.05
1.2
1.00 0.95 0.55
t = 0.05 Exact Suliciu (semi-disc. ent. satisf.) Suliciu (disc. ent. satisf.) Force GForce
0.60
0.65
0.70
0.75
1.0 0.55
0.80
(a) Density
0.60
0.65
0.70
0.75
0.80
(b) Discharge
Figure 6: Numerical simulations corresponding to (5.3). Comparison with original Suliciu scheme, Force and GForce schemes
31
1.25 1.20
t = 0.05 Exact Suliciu (semi-disc. ent. satisf.) Suliciu (disc. ent. satisf.) GForce+hyp. reconst.
2.0 1.8
t = 0.05 Exact Suliciu (semi-disc. ent. satisf.) Suliciu (disc. ent. satisf.) GForce+hyp. reconst.
1.15 1.6
1.10
1.4
1.05
1.2
1.00 0.95 0.55
0.60
0.65
0.70
0.75
1.0 0.55
0.80
(a) Density
0.60
0.65
0.70
0.75
0.80
(b) Discharge
Figure 7: Numerical simulations corresponding to (5.3). Comparison with High Order schemes known function which represents the topography. When z(x) is constant, the system reduces to a particular case of (3.1) and the scheme introduced in section 3 can be applied. In the more general case where z(x) is a given function, one may apply the hydrostatic reconstruction described in [1] to treat the source term. This hydrostatic reconstruction technique consist roughly in solving a Riemann problem by applying a given numerical flux for the homogeneous problem (with ∂x z = 0) to some reconstructed states related to the original left and right data. By applying this technique, one obtains a numerical flux that is semi-discrete entropy satisfying given that the homogeneous flux is semi-discrete entropy satisfying. Thus, a good choice for this homogenous flux would be the modified Suliciu scheme presented previously. We show a test taken from [7]. It is designed to assess the long time behavior and convergence to steady state. The space domain is [0, 25], and we introduce a topography given by ( 0.2 − 0.05(x − 10)2 if 8 < x < 12, z(x) = (5.5) 0 otherwise, with initial data ρ0 = 0.33,
u0 = 0.18/0.33.
(5.6)
The result is shown in Figure 8. We remark again that the semi-discrete approach is less diffusive in the shock region.
32
t=200
Semi−discrete entropy satisfying Discrete−entropy satisfying Exact solution
0.4
0.35
0.3
0.25
0.2
0.15
0.1
6
7
8
9
10
11
12
13
14
15
Figure 8: Density profile for (5.5)-(5.6)
6
Conclusion
The semi-discrete approach for an approximate Riemann problem lets to define stable schemes while it is less restrictive and allows to ask other properties like the exact resolution of stationary shocks. As a particular case, we have applied this technique to the Suliciu relaxation system and we have observed that not only the stationary shocks are preserved, but we obtain a slight amelioration for shocks. At the present stage, the schemes shown in sections 3 and 4 do not treat vacuum correctly in the sense that if one of the two densities ρl or ρr tends to 0 while the other remains finite, the propagation speed c/ρ will tend to infinity. Thus, the CFL condition would give a zero time-step for the scheme. Further work is being done by the authors in order to adapt the scheme to treat vacuum correctly.
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[3] F. Bouchut. Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math., 94(4):623–672, 2003. [4] F. Bouchut. Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkh¨ auser Verlag, Basel, 2004. [5] P. Colella and H. M. Glaz. Efficient solution algorithms for the Riemann problem for real gases. J. Comput. Phys., 59(2):264–289, 1985. [6] F. Coquel, E. Godlewski, B. Perthame, A. In, and P. Rascle. Some new Godunov and relaxation methods for two-phase flow problems. In Godunov methods (Oxford, 1999), pages 179–188. Kluwer/Plenum, New York, 2001. [7] T. Gallou¨et, J.-M. H´erard, and N. Seguin. Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. & Fluids, 32(4):479–513, 2003. [8] P. G. LeFloch. Hyperbolic systems of conservation laws. Lectures in Mathematics ETH Z¨ urich. Birkh¨auser Verlag, Basel, 2002. The theory of classical and nonclassical shock waves. [9] I. Suliciu. On modelling phase transitions by means of rate-type constitutive equations. Shock wave structure. Internat. J. Engrg. Sci., 28(8):829–841, 1990. [10] I. Suliciu. Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations. Internat. J. Engrg. Sci., 30(4):483–494, 1992. [11] E. F. Toro. Riemann solvers and numerical methods for fluid dynamics. Springer-Verlag, Berlin, second edition, 1999. A practical introduction. [12] E. Toro, and V. A. Titarev MUSTA fluxes for systems of conservation laws. Journal of Computational Physics, 216(2):403–429, Aug. 2006. [13] A. Marquina. Local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws. SIAM Journal on Scientific Computing, 15(4):892–915, July 1994.
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