Numerical schemes for Euler equations: a physically admissible extension to order 2
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(2) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Issue § System of conservation laws modelling a physical phenomenon like in fluid mechanics, ... ( @t W + r F (W) = 0; W(0; x) = W0 (x): § Physical constraints: W 2 W à Maximum principle (Euler for incompressible fluids: density) à Positivity (Euler for compressible fluids: density and pressure). § These constraints must be satisfied: à at the continuous level (relevance of the mathematical model) à at the discrete level (robustness of the numerical scheme). 2 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(3) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Example Euler equations for a 2D perfect fluid: W = t (; u; E );. W=. . W 2 R4 :. F = t (u; u. = W1 > 0. . et p = (. (. Riemann problem: W0 (x) =. u + p I2 ; (E + p)u). Wl ; Wr ;. if x1 if x1. W22 + W32 2W1. 1) W4. . . >0. < 0; > 0:. Rarefaction waves and vacuum (Einfeldt, Munz, Roe & Sjögreen, 1991): § §. 0 > 0, u0 > 0, E0 > u02 =2 Wl = (0 ; 0 u0 ; 0; 0 E0 ) and Wr. § If. 4 3. 1. E0. = (0 ; 0 u0 ; 0; 0 E0 ). > u02 , then density and pressure remain positive. 3 / 19. Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(4) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Positivity-preserving schemes 1st order § Einfeldt et al. (1991), Bouchut (2004) § Godunov, Rusanov, HLLs / Roe 2nd order: FV schemes + MUSCL strategy § Scalar equations: Clain & Clauzon (2010), Calgaro et al. (2010) à Modification of limiters à Adaptation of the CFL condition. § Systems of conservation laws: à . . . monoslope reconstruction: Perthame & Shu (1996) à . . . multislope reconstruction: Berthon (2006). W. Method based on the convexity of Adaptation of the reconstruction procedure Addition of a fictitious state in each cell Reformulation of the 2nd order 2D scheme as a convex combination of 1st order 1D schemes. 4 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(5) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Positivity-preserving schemes 1st order § Einfeldt et al. (1991), Bouchut (2004) § Godunov, Rusanov, HLLs / Roe 2nd order: FV schemes + MUSCL strategy § Scalar equations: Clain & Clauzon (2010), Calgaro et al. (2010) à Modification of limiters à Adaptation of the CFL condition. § Systems of conservation laws: à . . . monoslope reconstruction: Perthame & Shu (1996) à . . . multislope reconstruction: Berthon (2006). W. Method based on the convexity of Adaptation of the reconstruction procedure Addition of a fictitious state in each cell Reformulation of the 2nd order 2D scheme as a convex combination of 1st order 1D schemes. 4 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(6) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Finite volume schemes: 1st-order Mq Ωi Ωr. Ml Γil. Mk. Mr nil. Mp. Mi. Mm. Wn+1 = Wni i. ∆t n. jΓij j F j Ωi j j 2V (i) X. Wni ; Wnj ; nij. . 5 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(7) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Finite volume schemes: 2nd-order (MUSCL) Mq Ωi Ωr. Ml. Mk. Mr nil. Γil. Mp. Mi. Mm. Wn+1 = Wni i. ∆t n. jΓij j F jΩi j j 2V (i) X. Wnij ; Wnji ; nij. . 5 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(8) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Modification of Berthon’s strategy (2006) Wn+1 = Wni i. ∆t n. X jΓij j n n jΩi j F Wij ; Wji ; nij. j 2V (i). 6 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(9) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Modification of Berthon’s strategy (2006) Wn+1 = Wni i. ∆t n. X jΓij j n n jΩi j F Wij ; Wji ; nij. j 2V (i). Wni =. jΩi j W + jΩi j i. X jΩij j n jΩi j Wij. j 2V (i). 6 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(10) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Modification of Berthon’s strategy (2006) Wn+1 = Wni i. ∆t n. X jΓij j n n jΩi j F Wij ; Wji ; nij. j 2V (i). Wni =. jΩi j W + jΩi j i. X jΩij j n jΩi j Wij. j 2V (i). Wn+1 = i. jΩi j W + jΩi j i. X jΩij j jΩi j Wij. j 2V (i). 6 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(11) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Modification of Berthon’s strategy (2006) Wn+1 = Wni i. ∆t n. X jΓij j n n jΩi j F Wij ; Wji ; nij. j 2V (i). Wni =. jΩi j W + jΩi j i. X jΩij j n jΩi j Wij. j 2V (i). Wn+1 = i. W = Wn. ∆t F (Wn ; Vn ; n). `. for a suitable 1D flux. jΩi j W + jΩi j i. X jΩij j jΩi j Wij. j 2V (i). F (Wn ; Wn ; n). . F and a small enough time step ∆t 6 / 19. Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(12) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Modification of Berthon’s strategy (2006) Wn+1 = Wni i. ∆t n. X jΓij j n n jΩi j F Wij ; Wji ; nij. j 2V (i). Wni =. i Wi. i ). + (1. X. ij Wnij. j 2V (i). Wn+1 = i. X. i Wi + (1 i ). ij Wij. j 2V (i). W = Wn. ∆t F (Wn ; Vn ; n). `. for a suitable 1D flux. F (Wn ; Wn ; n). . F and a small enough time step ∆t 6 / 19. Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(13) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Modification of Berthon’s strategy (2006) How to choose i and. ij ?. 6 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(14) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Modification of Berthon’s strategy (2006) How to choose i and. ij ?. Accuracy. Computation of the additional state Wi 2 W “large” i. 6 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(15) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Modification of Berthon’s strategy (2006) How to choose i and. ij ?. Accuracy. Efficiency. Computation of the additional state Wi 2 W. Optimisation of the CFL condition. “large” i. “small” i. 6 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(16) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Modification of Berthon’s strategy (2006) How to choose i and. ij ?. Accuracy. Efficiency. Computation of the additional state Wi 2 W. Optimisation of the CFL condition. “large” i. “small” i. Balance?. 6 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(17) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Outline. 1. Introduction. 2. Admissibility. 3. Computation of the time step. 4. Simulations. 7 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(18) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Reconstruction across interfaces Issue: ensure that Wij. 2W Mq Downstream triangle Upstream triangle Ml. Mk. Qij. Mj. Mp. Mi Mm. 8 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(19) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Reconstruction across interfaces Issue: ensure that Wij. 2W Mq Downstream triangle Upstream triangle Ml. Mk. Mj. Qij. Mp. Mi Mm. Reconstruction of physical variables U = (; u; p) = (W) ∆ ij. up. ij = i +. ij '(rij )(j. i );. rij =. j. i 8 / 19. Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(20) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Reconstruction across interfaces Issue: ensure that Wij. 2W Mq Downstream triangle Upstream triangle Ml. Mk. Qij. Mj. Mp. Mi Mm. Use of -limiters ij. =. Mi Qij ; Mi Mj. . = min i ;j. 1 ij. 2 [1; 2];. 0 6 '(r ) 6 min( r ; ). 8 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(21) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Reconstruction across interfaces Issue: ensure that Wij. 2W Mq Downstream triangle Upstream triangle Ml. Mk. Qij. Mj. Mp. Mi Mm. Gradient with conservative variables Wij = Wi + ∆Wij ;. ∆Wij = . 1. Ui + ∆Uij. . Wi. 8 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(22) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Key-point of the procedure The additional state must be physically admissible Wnij = Wni + ∆Wnij ; 2. 1 Wi = 4Wni. i. (1. X. i ) j. 2V (i). 3. ij Wnij 5 = Wni. 1. i. i. X j. 2V (i). ij ∆Wnij. 9 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(23) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Key-point of the procedure The additional state must be physically admissible Wnij = Wni + 2. n n i ∆Wij. 1 Wi = 4Wni. i. n i. ;. (1. i ). X. n i j. 2 [0; 1] =). 2V (i). Wnij = (1. 3. 1. ij Wnij 5 = Wni. i. i. n n i )Wi. +. n i. X. n i j. . Wni + ∆Wnij. 2V (i). . ij ∆Wnij. 2W. 9 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(24) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Key-point of the procedure The additional state must be physically admissible Wnij = Wni + 2. n n i ∆Wij. 1 Wi = 4Wni. i. n i. ; i ). (1. X. n i j. 2 [0; 1] =). 2V (i). 3. Wnij = (1. Let us set. i. . n n i )Wi. X j. 2V (i). +. n i. X. n i. i. ∆Wi =. 1 i so that Wi = Wni + . 1. ij Wnij 5 = Wni. j. . Wni + ∆Wnij. 2V (i). . ij ∆Wnij. 2W. ij ∆Wnij. .. n i ∆Wi. i. Choice of. n i. prescribed by. i > 0 and pi > 0 9 / 19. Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(25) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Choice of. n i. i > 0 () P1. . . 1. 1 p > 0 () P2 i. Density Given Din =. n i. () i. 6. Pressure. n i. 6. . i. i i. ni. . := min 1;. with. . := 1 +. Bin. 1. 1. . i. i i. i. n i. >0. n i. . +. Ani. i. 1. i. 2 n i. >0. , we derive a first condition:. . (p) i. . := 1 + Din. n i. i. ∆i. n i. i. 1. i. ( ) i. . with. () i. =. 8 > <+. 1;. if Din. > :. 1 ; Din. otherwise.. 0;. ::: 10 / 19. Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(26) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Convex combinations n+1. Wi. =. n. Wi. ∆t. n. X j 2V (i). n Wi = i Wi + (1. i ). X. ij Wnij. jΓij j F jΩi j n+1. Wi. n. n. Wij ; Wji ; nij. = i Wi + (1. . i ). j 2V (i). X. ij Wij. j 2V (i). 11 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(27) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Convex combinations n+1. Wi. n. =. Wi. ∆t. n. X j 2V (i). n Wi = i Wi + (1. i ). X. ij Wnij. jΓij j F jΩi j n+1. Wi. n. n. Wij ; Wji ; nij. . = i Wi + (1. i ). j 2V (i). Wij. 8 > <Wij > :. =. n. Wij. 4 X. =. ∆t. n. X. ij Wij. j 2V (i). 4 X. . ij ;k F Wnij ; Wnij ;k ; nij ;k ;. 2 V (i). j. k=1. ij ;k Wij ;k ij ;k. k=1 n. Wij ;k = Wij. n. . n. n. ∆t ij ;k F (Wij ; Wij ;k ; nij ;k ). n. n. F (Wij ; Wij ; nij ;k ). . 11 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(28) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Assumptions Flux In addition to classical properties, we assume. 8 (V ; W ) 2 W 2 ; W. ∆t. `. [F (W ; V ). F (W ; W )] 2 W ;. under the CFL condition ∆t max jk (V ; W )j 6 k. 0. `:. CFL Condition ∆t n max j. 2V (i). . ij ;. . max. . 1 k 4. ij ;k ¯ ni ¯ ni :=. 0. max. 2V . j (i) 1 k 4. . junij nij ;k j + cijn ; jui nij ;k j + ci. Aim Choose i and (ij ) such that ∆t n is maximal, i.e. maxfij ; ij ;k g are minimal.. ¯ n and such that i. 12 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(29) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Constraints Sign § §. ij ;k 0, ij 0 ij ;k 0. Consistency of the decomposition § § § § §. ij ij ;2 = ik ik ;4 (1 i )ij ij ;1 = i ij jΓ j (1 i )ij ij ;3 = jΩ j 4 4 X X ij ij ;k = 1, =1 ij ;k k=1 k=1 ij ij i. 4 X. ij ;k nij ;k = 0. k=1. 13 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(30) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Resolution. Xmax := (1 jΓij)j jΩ j for some j0 2 V (i), we have ij i i jΓ j jΓ j (1 i )ij X ; = jΓij j ij ;1 = ij ij X ; ij = ij ij ;3 jΓij j ij jΓij j i (1 i )ij jΩi j. With X = ij0 ;1. 0. 0. 0. 0. 0. ij ;2 = ij ;k =. Mi Gijl. ij. 0. . 1. i )jΩi j. (1. j@ Tij j. (1 i )ij jΩi j opt. i. . ij. jΓij j. X ij0 jΓij0 j ij. 0. X. ; ij ;4 =. Mi Gijk. ij. 0. j@ Tij j. 2jΓij j. . ;. . 1. i )jΩi j. (1. ij =. . ij. jΓij j 0. X. 0. j@ Ωi j (1 i )ij jΓij j i. 0. X. 0. . . := min max ij (X ); max ij ;k (X ) : 0X Xmax j 2V (i) 1k 4 14 / 19. Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(31) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Resolution. Optimisation The optimal coefficient reads 8 > > > > < (1. opt i (i ; ij ) = > > > > :. 2. jΓij j ;. i )jΩi j j 2V (i) ij max. if. i i ;. j@ Ωi j min 1 2jΓij j ij j@ Ωi j + ij j@ Ωi j jΩi j j 2V (i) i j@ Tij j j@ Tij j j@ Tij j. 1. An optimal bound for the solution is given by opt. i. . 1 jΓ j i = ; ij = ij 3. . j@ Ωi j. =3. j@ Ωi j : jΩi j. 14 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(32) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Example. 15 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(33) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Algorithm 1 2. Iteration 0: computation of ij and Iterations n > 1: computation of à local gradients ∆Wnij à intermediate gradient ∆Wi = à. n i. = minf1;. 1 2. opt i P. ij ∆Wnij. j 2V (i). #(i ) ; 21 #i(p) g. à reconstructed states Wnij = Wni + Wi = Wni. n n i ∆Wij. 2. and. X. n i. ij ∆Wnij. j 2V (i). à eigenvalues and time step ∆t n =. 0. opt ¯ ni i. à updated state Wn+1 i 16 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(34) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. 1-2-3 test case. 17 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(35) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. 1-2-3 test case. 17 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(36) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. 1-2-3 test case. 17 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(37) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. 1-2-3 test case. 17 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
(38) 1. Introduction 2. Admissibility 3. CFL 4. Simulations 5. Conclusion. Perspectives Done X Analysis of robustness of general MUSCL strategies X Derivation of (sufficient) conditions to preserve positivity and entropy inequalities X Explicit CFL conditions X Direct extension to 2nd-order in time (RK2) X Easy adaptation of industrial codes C. Calgaro, E. Creusé, T. Goudon & Y. Penel, Positivity-preserving schemes for Euler equations: sharp and practical CFL conditions (J. Comput. Phys., 234, 2013). To do § Influence of the numerical flux on. n i. § Application to other systems of conservation laws 18 / 19 Y. Penel (CEREMA). MUSCL schemes for Euler. /. /. -. :.
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