Finite volume method for conservation laws V

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1 Finite volume method for conservation laws V Schemes satisfying entropy condition Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore February 1, / 18

2 Entropy condition Let U(u) be any convex entropy of the scalar conservation law with associated entropy flux F (u), i.e., u t + x f(u) = 0 U (u)f (u) = F (u) Then the unique entropy solution of the conservation law satisfies in the sense of distributions U t + x F (u) 0 / 18

3 Definition (Entropy consistent numerical scheme) The difference scheme H is said to be consistent with the entropy condition if there exists a continuous function G : R k R which satisfies the following requirements (1) consistency with the entropy flux F () cell entropy inequality G(v,..., v) = F (v) U(v n+1 j ) U(vj n) + G(vn j k+1,..., vn j+k ) G(vn j k,..., vn j+k 1 ) 0 t x The function G is called the numerical entropy flux. If we set U n j = U(v n j ), G n j+ 1 = G(v n j k+1,..., v n j+k) then we have to verify that U n+1 j Uj n λ(g n j+ G n 1 j ) 1 3 / 18

4 When the scheme is consistent with any entropy function, we say that it is an entropy scheme. Proposition 4.1 (GR1, Chap. 3) Let H be a 3-point difference scheme with C numerical flux which is consistent with any entropy condition. Then it is at most first order accurate. Theorem Assume that the hypothesis of Lax-Wendroff theorem hold. Assume moreover that the scheme is consistent with any entropy condition. Then the limit u is the unique entropy solution of the conservation law. Theorem A monotone consistent scheme is consistent with any entropy condition. Proof: It is enough to check the entropy condition for the Kruzkov s entropy functions: for any l R U(u) = u l, F (u) = sign(u l)(f(u) f(l)) 4 / 18

5 Set a b = min(a, b), a b = max(a, b) and define the numerical entropy flux G as follows G(v k+1,..., v k ) = g(v k+1 l,..., v k l) g(v k+1 l,..., v k l) We first prove the following identity v j l λ(g j+ 1 G j 1 ) = H(v j k l,..., v j+k l) H(v j k l,..., v j+k l) By definition of G j+ 1 G j+ 1 G j 1 = [g(v j k+1 l,..., v j+k l) g(v j k+1 l,..., v j+k l)] [g(v j k l,..., v j+k 1 l) g(v j k l,..., v j+k 1 l)] = [g(v j k+1 l,..., v j+k l) g(v j k l,..., v j+k 1 l)] [g(v j k+1 l,..., v j+k l) g(v j k l,..., v j+k 1 l)] and using the finite volume scheme, we get λ[g j+ 1 j 1 ] = [H(v j k l,..., v j+k l) v j l] [H(v j k l,..., v j+k l) v j l] 5 / 18

6 which gives the identity we wanted to show since v j l v j l = v j l Now since H is monotone and consistent H(v j k l,..., v j+k l) H(v j k,..., v j+k ) H(l,..., l) = v n+1 j In the same way we obtain H(v j k l,..., v j+k l) v n+1 j Combining the above results we verify the entropy condition l v n+1 j l vj n l + λ(g n j+ G n 1 j ) 0 1 The consistency condition of G is readily satisfied since G(v,..., v) = g(v l,..., v l) g(v l,..., v l) = f(v l) f(v l) = sign(v l)(f(v) f(l)) = F (u) l 6 / 18

7 Definition (E-scheme) A consistent, conservative scheme is called an E-scheme if its numerical flux satisfies sign(v j+1 v j )(g j+ 1 f(u)) 0 for all u between v j and v j+1. Remark: Note that an E-scheme is essentially 3-point. Indeed letting v j+1 v j with first v j+1 v j and then with v j+1 v j shows that g is essentially 3-point. Remark: A 3-point monotone scheme is an E-scheme. Since g(u, v) is non-decreasing in u and non-increasing in v, we obtain and therefore g(u, v) g(u, w) g(w, w) = f(w) if u w v g(u, v) g(w, v) g(w, w) = f(w) if u w v sign(v u)(g(u, v) f(w)) 0, for all w between u and v In particular, the Godunov scheme is an E-scheme under CFl 1. 7 / 18

8 Lemma Assume that CFL 1. Then E-fluxes are characterized by { gj+ 1 gg j+ 1 g j+ 1 gg j+ 1 if v j < v j+1 if v j > v j+1 where g G stands for Godunov numerical flux. Proof: Under CFL 1, the Godunov flux is given by { g G min u [vj,v j+ = j+1] f(u) if v j < v j+1 1 max u [vj+1,v j] f(u) if v j v j+1 Assume v j < v j+1. Then E-flux satisfies g j+ 1 f(u), v j u v j+1 Assume v j > v j+1. Then E-flux satisfies v j+1 u v j g j+ 1 f(u), 8 / 18

9 Lemma Assume that CFL 1. E-schemes are characterized by 0 Q G j+ 1 Q j+ 1, j Z Proof: Assume v j < v j+1. Then for E-scheme g j+ 1 gg j+ 1 1 (f j + f j+1 ) 1 Q j+ 1 (v j+1 v j ) 1 (f j + f j+1 ) 1 QG j+ (v 1 j+1 v j ) Proposition An E-scheme with differentiable numerical flux is at most first order accurate. 9 / 18

10 Theorem (Viscous form and entropy condition) Assume that the CFL condition λ max a(u) 1 holds. An E-scheme whose coefficient of numerical viscosity satisfies Q G j+ 1 Q j+ 1 1 is consistent with any entropy condition. The proof requires two lemmas which we first prove. The basic idea is to write any E-scheme as a convex combination of the Godunov scheme and a modified Lax-Friedrichs scheme, both of which satisfy entropy condition. 10 / 18

11 Lemma (Godunov scheme) Assume that CFL 1. The Godunov scheme can be written in the following way where H G (v j 1, v j, v j+1 ) = 1 (vg + v G+ ) j 1 j+ 1 v G j 1 v G+ j+ 1 = = x x x 0 0 x w R (x/ t; v j 1, v j )dx = v j λ(f j g G j 1 ) w R (x/ t; v j, v j+1 )dx = v j + λ(f j g G j+ 1 ) Moreover if (U, F ) is any entropy pair, we have U(v G± ) U(v j± 1 j ) ± λ[f (v j ) G G j± ] 1 where G G (u, v) = F (w R (0; u, v)) 11 / 18

12 Proof: The formulae for v G± follow from the derivation of the Godunov j± 1 scheme. Now since U is a convex function, we obtain by Jensen s inequality U ( x x 0 w R (x/ t; v j 1, v j )dx ) x U(w R (x/ t; v j 1, v j ))dx x 0 Since w R (x/ t; v j 1, v j ) is by definition the entropy solution of the Riemann problem u t + x u(x, 0) = it satisfies an entropy inequality U(u) t f(u) = 0, { v j 1 x < 0 v j x > 0 + F (u) x t [0, t] 0 1 / 18

13 Integrating this last inequality on the domain (x j 1, x j) (0, t) we get In the same way we have x U(v G ) U(w j 1 R (x/ t; v j 1, v j ))dx x 0 U(v j ) λ[f (v j ) G G j ] 1 U(v G+ 0 ) U(w j+ 1 R (x/ t; v j, v j+1 ))dx x x U(v j ) + λ[f (v j ) G G j+ ] 1 Remark: This also shows that Godunov scheme satisfies the entropy condition associated with (U, F ) since U(H G (v j 1, v j, v j+1 )) 1 [U(vG ) + U(v G+ )] j 1 j+ 1 U(v j ) λ[g G j+ G G 1 j ] 1 13 / 18

14 Lemma (Lax-Friedrichs modified) Consider the 3-point scheme H M (v j 1, v j, v j+1 ) = 1 4 (v j 1 + v j + v j+1 ) 1 λ(f j+1 f j 1 ) Then under the condition CFL 1, we can write where v M j 1 v M+ j+ 1 = = H M (v j 1, v j, v j+1 ) = 1 (vm j 1 + v M+ j+ 1 ) x 1 w R (x/ t; v j 1, v j )dx = 1 x x (v j + v j 1 ) λ(f j f j 1 ) 1 x x x Moreover if (U, F ) is any entropy pair, we have where w R (x/ t; v j, v j+1 )dx = 1 (v j + v j+1 ) λ(f j+1 f j ) U(v M± j± 1 ) U(v j ) ± λ[f (v j ) G M j± 1 ] G M (u, v) = 1 1 (F (u) + F (v)) (U(v) U(u)) 4λ 14 / 18

15 Proof: Under the condition CFL 1, we obtain the desired results by integrating the conservation law over the domains (x j 1, x j ) (t n, t n+1 ) and (x j, x j+1 ) (t n, t n+1 ). We get 0 = = tn+1 xj t n x x t 0 x j 1 ( wr t + ) x f(w R) dxdt w R (x/ t; v j 1, v j )dx x (v j + v j 1 ) + [f(w R ( x/(t); v j 1, v j )) f(w R ( x/(t); v j 1, v j ))]dt But the waves from x j 1 do not reach x j 1, x j so that w R ( x/(t); v j 1, v j ) = v j, w R ( x/(t); v j 1, v j ) = v j 1 This proves the formulae for v M±. j± 1 15 / 18

16 Since w R is the entropy solution, we integrate the entropy inequality over (x j 1, x j ) (t n, t n+1 ) and using Jensen s inequality U(v M ) j 1 x 1 U(w R (x/ t; v j 1, v j ))dx x x 1 (U(v j) + U(v j 1 )) λ(f (v j ) F (v j 1 )) = U(v j ) λ(f (v j ) G M j 1 ) Remark: The modified Lax-Friedrichs scheme is consistent with any entropy condition with numerical flux G M since U(H M (v j 1, v j, v j+1 )) 1 (U(vM ) + U(v M+ )) j 1 j+ 1 U(v j ) λ[g M j+ G M 1 j ] 1 Note that the numerical viscosity of this scheme is Q M j+ 1 = 1 16 / 18

17 Proof of Theorem (Viscous form and entropy condition): Let us write the scheme in viscous form v n+1 j = vj n λ (f j+1 n fj 1) n + 1 (Qn j+ v n 1 j+ Q n 1 j v n 1 j ) 1 and in averaged form where v n+1 j = 1 (v + v + ) j 1 j+ 1 v j 1 v + j+ 1 = v j λ(f j f j 1 ) Q j 1 v j 1 = v j λ(f j+1 f j ) + Q j+ 1 v j+ 1 We can write the Godunov and modified Lax-Friedrich schemes in the same form with superscript G and M. Now since Q M = 1 and Q G Q Q M = 1 we can write, with some 0 θ j / 18

18 It follows that Q j+ 1 = θ j+ 1 QG j+ + (1 θ 1 j+ 1 )QM j+, j Z 1 v ± j± 1 = θ j± 1 vg± + (1 θ j± 1 j± 1 )vm± j± 1 If (U, F ) is any entropy pair, then due to convexity of U where U(v n+1 j ) 1 U(v j 1 ) + 1 U(v+ j+ 1 ) 1 θ j 1 U(vG ) + 1 j 1 (1 θ j 1 )U(vM ) + j 1 1 θ j+ 1 U(vG+ j+ 1 U(v j ) λ(g j+ 1 G j 1 ) ) + 1 (1 θ j+ 1 )U(vM+ ) j+ 1 G j+ 1 = θ j+ 1 GG j+ + (1 θ 1 j+ 1 )GM j+ 1 is a consistent entropy flux associated with the E-scheme under consideration. Remark: Under the conditions of the above theorem Q G Q 1, the E-scheme is also TVD and L stable. 18 / 18

Math 660-Lecture 23: Gudonov s method and some theories for FVM schemes

Math 660-Lecture 23: Gudonov s method and some theories for FVM schemes Math 660-Lecture 3: Gudonov s method and some theories for FVM schemes 1 The idea of FVM (You can refer to Chapter 4 in the book Finite volume methods for hyperbolic problems ) Consider the box [x 1/,