Practical CFL conditions for MUSCL schemes solving Euler equations Yohan Penel 1 1 LRC Manon – UPMC-LJLL, Univ. Paris 6 Joint work with C. Calgaro & E. Creus´ e (Univ. Lille 1, INRIA Lille) T. Goudon (INRIA Sophia-Antipolis) 14th International Conference on Hyperbolic Problems: Theory, Numerics, Applications Padova – June, 25th. 2012
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Outline Introduction 1 Computation of the time step 2 Admissibility 3 Simulations 4 2 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 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 ) = W 0 ( x ) ✿ ❧ Physical constraints : W ✷ ❲ ➠ Maximum principle (Euler for incompressible fluids: density) ➠ Positivity (Euler for compressible fluids: density and pressure) ❧ These constraints must be satisfied: ➠ at the continuous level (relevance of the mathematical model) ➠ at the discrete level (robustness of the numerical scheme) 3 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Example Euler equations for a 2D perfect fluid: W = t ( ✚❀ ✚ u ❀ ✚ E ) ❀ F = t ( ✚ u ❀ ✚ u ✡ u + p ■ 2 ❀ ( ✚ E + p ) u ) W 4 � W 2 2 + W 2 ✚ ✔ ✕ ✛ W ✷ R 4 : ✚ = W 1 ❃ 0 et p = ( ✌ � 1) 3 ❲ = ❃ 0 2 W 1 ✭ W l ❀ if x 1 ❁ 0 ❀ Riemann problem : W 0 ( x ) = W r ❀ if x 1 ❃ 0 ✿ Rarefaction waves and vacuum (Einfeldt, Munz, Roe & Sjgreen, 1991): ❧ ✚ 0 ❃ 0, u 0 ❃ 0, E 0 ❃ u 2 0 ❂ 2 ❧ W l = ( ✚ 0 ❀ � ✚ 0 u 0 ❀ 0 ❀ ✚ 0 E 0 ) and W r = ( ✚ 0 ❀ ✚ 0 u 0 ❀ 0 ❀ ✚ 0 E 0 ) 4 ✌ 3 ✌ � 1 E 0 ❃ u 2 ❧ If 0 , then density and pressure remain positive . 4 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 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) ➠ Modification of limiters ➠ Adaptation of the CFL condition ❧ Systems of conservation laws : ➠ . . . monoslope reconstruction: Perthame & Shu (1996) ➠ . . . multislope reconstruction: Berthon (2006) 5 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion MUSCL strategy M q Ω i Ω j M l M j n il Γ il M p M k M i M m ❥ Γ ij ❥ W n +1 = W n i � ∆ t n ❳ W n ij ❀ W n � ✁ ji ❀ n ij ❥ Ω i ❥ F i j ✷❱ ( i ) 6 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Modification of Berthon’s strategy (2006) ❥ Γ ij ❥ i � ∆ t n ❳ W n +1 W n ❥ Ω i ❥ F � W n ij ❀ W n ✁ = ji ❀ n ij i j ✷❱ ( i ) 7 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Modification of Berthon’s strategy (2006) ❥ Γ ij ❥ i � ∆ t n ❳ W n +1 W n ❥ Ω i ❥ F � W n ij ❀ W n ✁ = ji ❀ n ij i j ✷❱ ( i ) ❥ Ω ✄ i ❥ ❥ Ω ij ❥ ❳ W n ❥ Ω i ❥ W ✄ ❥ Ω i ❥ W n = i + i ij j ✷❱ ( i ) 7 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Modification of Berthon’s strategy (2006) ❥ Γ ij ❥ i � ∆ t n ❳ W n +1 W n ❥ Ω i ❥ F � W n ij ❀ W n ✁ = ji ❀ n ij i j ✷❱ ( i ) ❥ Ω ✄ i ❥ ❥ Ω ij ❥ ❳ W n ❥ Ω i ❥ W ✄ ❥ Ω i ❥ W n = i + i ij j ✷❱ ( i ) ❥ Ω ✄ i ❥ ❥ Ω ij ❥ ✄ ❳ W n +1 = i + ❥ Ω i ❥ W ❥ Ω i ❥ W ij i j ✷❱ ( i ) 7 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Modification of Berthon’s strategy (2006) ❥ Γ ij ❥ i � ∆ t n ❳ W n +1 W n ❥ Ω i ❥ F � W n ij ❀ W n ✁ = ji ❀ n ij i j ✷❱ ( i ) ❥ Ω ✄ i ❥ ❥ Ω ij ❥ ❳ W n ❥ Ω i ❥ W ✄ ❥ Ω i ❥ W n = i + i ij j ✷❱ ( i ) ❥ Ω ✄ i ❥ ❥ Ω ij ❥ ✄ ❳ W n +1 = i + ❥ Ω i ❥ W ❥ Ω i ❥ W ij i j ✷❱ ( i ) W n � ∆ t F ( W n ❀ V n ❀ n ) � F ( W n ❀ W n ❀ n ) ✄ ✂ W = ❵ for a suitable 1D flux ❋ and a small enough time step ∆ t 7 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Modification of Berthon’s strategy (2006) ❥ Γ ij ❥ i � ∆ t n ❳ W n +1 W n ❥ Ω i ❥ F � W n ij ❀ W n ✁ = ji ❀ n ij i j ✷❱ ( i ) ❳ W n = ✑ ✄ i W ✄ i + (1 � ✑ ✄ ✑ ij W n i ) i ij j ✷❱ ( i ) ✄ ❳ W n +1 = ✑ ✄ i + (1 � ✑ ✄ i ) i W ✑ ij W ij i j ✷❱ ( i ) W n � ∆ t F ( W n ❀ V n ❀ n ) � F ( W n ❀ W n ❀ n ) ✄ ✂ W = ❵ for a suitable 1D flux ❋ and a small enough time step ∆ t 7 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Modification of Berthon’s strategy (2006) How to choose ✑ ✄ i and ✑ ij ? Accuracy Efficiency Computation of Optimization of the additional the CFL condition state W ✄ ✷ ❲ i ”large” ✑ ✄ ”small” ✑ ✄ Balance? i i 7 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Outline Introduction 1 Computation of the time step 2 Admissibility 3 Simulations 4 8 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Outline Introduction 1 Computation of the time step 2 Admissibility 3 Simulations 4 9 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Convex combinations ❥ Γ ij ❥ � ∆ t n ❳ ❥ Ω i ❥ F � ✁ W n +1 W n W n ij ❀ W n = ji ❀ n ij i i j ✷❱ ( i ) ❳ ❳ ✄ W n i = ✑ ✄ i W ✄ i + (1 � ✑ ✄ ✑ ij W n W n +1 = ✑ ✄ i + (1 � ✑ ✄ i ) i W i ) ✑ ij W ij ij i j ✷❱ ( i ) j ✷❱ ( i ) 10 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Convex combinations ❥ Γ ij ❥ � ∆ t n ❳ ❥ Ω i ❥ F � ✁ W n +1 W n W n ij ❀ W n = ji ❀ n ij i i j ✷❱ ( i ) ❳ ❳ ✄ W n i = ✑ ✄ i W ✄ i + (1 � ✑ ✄ ✑ ij W n W n +1 = ✑ ✄ i + (1 � ✑ ✄ i ) i W i ) ✑ ij W ij ij i j ✷❱ ( i ) j ✷❱ ( i ) 4 ❳ W n ij � ∆ t n ✏ ij ❀ k F � W n ij ❀ W n ✁ = ij ❀ k ❀ n ij ❀ k j ✷ ❱ ( i ) W ij ❀ k =1 4 ✽ ✏ ij ❀ k ❳ ❃ W ij = W ij ❀ k ❁ ✖ ij ❀ k k =1 ❃ W ij ❀ k = W n ij � ∆ t n ✖ ij ❀ k ✂ F ( W n ij ❀ W n ij ❀ k ❀ n ij ❀ k ) � F ( W n ij ❀ W n ij ❀ n ij ❀ k ) ✄ ✿ 10 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Assumptions Flux In addition to classical properties, we assume: ✽ ( V ❀ W ) ✷ ❲ 2 ❀ W � ∆ t ❵ [ ❋ ( W ❀ V ) � ❋ ( W ❀ W )] ✷ ❲ under the CFL condition ∆ t max ❥ ✕ k ( V ❀ W ) ❥ � ☛ 0 ❵ . k ✚ ✛ CFL Condition ∆ t n ✂ max ✖ ✄ ✂ ¯ ✕ n ij ❀ max 1 ✔ k ✔ 4 ✖ ij ❀ k i ✔ ☛ 0 j ✷❱ ( i ) ¯ ✕ n ❥ u n ij ✁ n ij ❀ k ❥ + c n ij ❀ ❥ u n ij ❀ k ✁ n ij ❀ k ❥ + c n ✟ ✠ i := max ij ❀ k j ✷❱ ( i ) 1 ✔ k ✔ 4 Optimization The optimal coefficient reads ✽ 2 ❥ Γ ij ❥ si ✑ ✄ i � ✑ ✄ i ) ❥ Ω i ❥ max ❀ ❃ ❃ i (1 � ✑ ✄ ✑ ij ❃ j ✷❱ ( i ) ❃ ❁ ✖ ♦♣t ( ✑ ✄ i ❀ ✑ ij ) = i ✛✕ � 1 ❥ ❅ Ω i ❥ ✔ ✚ ✒ 1 � 2 ❥ Γ ij ❥ ❥ ❅ ❚ ij ❥ � ✑ ij ❥ ❅ Ω i ❥ ✓ + ✑ ij ❥ ❅ Ω i ❥ ❃ ✑ ✄ ❃ min ❃ ❃ i ❥ Ω i ❥ ❥ ❅ ❚ ij ❥ ❥ ❅ ❚ ij ❥ ✿ j ✷❱ ( i ) 11 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Assumptions Flux In addition to classical properties, we assume: ✽ ( V ❀ W ) ✷ ❲ 2 ❀ W � ∆ t ❵ [ ❋ ( W ❀ V ) � ❋ ( W ❀ W )] ✷ ❲ under the CFL condition ∆ t max ❥ ✕ k ( V ❀ W ) ❥ � ☛ 0 ❵ . k ✚ ✛ CFL Condition ∆ t n ✂ max ✖ ✄ ✂ ¯ ✕ n ij ❀ max 1 ✔ k ✔ 4 ✖ ij ❀ k i ✔ ☛ 0 j ✷❱ ( i ) ¯ ✕ n ❥ u n ij ✁ n ij ❀ k ❥ + c n ij ❀ ❥ u n ij ❀ k ✁ n ij ❀ k ❥ + c n ✟ ✠ i := max ij ❀ k j ✷❱ ( i ) 1 ✔ k ✔ 4 Optimization An optimal bound for the solution is given by: ✒ i = 1 3 ❀ ✑ ij = ❥ Γ ij ❥ ✓ = 3 ❥ ❅ Ω i ❥ ✖ ♦♣t ✑ ✄ ❥ Ω i ❥ ✿ i ❥ ❅ Ω i ❥ 11 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
1. Introduction 2. CFL 3. Admissibility 4. Simulations 5. Conclusion Example 12 / 17 Y. Penel (LJLL) Positive schemes for Euler / / - :
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