Practical CFL conditions for MUSCL schemes solving Euler equations - - PowerPoint PPT Presentation

Practical CFL conditions for MUSCL schemes solving Euler equations

Yohan Penel1

1LRC 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

1

Introduction

2

Computation of the time step

3

Admissibility

4

Simulations

• Y. Penel (LJLL)

Positive schemes for Euler

• 2 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Issue

❧ System of conservation laws modelling a physical phenomenon like in fluid mechanics, ... ✭ ❅tW + r ✁ F(W) = 0❀ W(0❀ x) = W0(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)

• Y. Penel (LJLL)

Positive schemes for Euler

• 3 / 17

/ / :

• 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 ✷ R4 : ✚ = W1 ❃ 0 et p = (✌ 1) ✔ W4 W 2

2 + W 2 3

2W1 ✕ ❃ 0 ✛ Riemann problem: W0(x) = ✭ Wl❀ if x1 ❁ 0❀ Wr❀ if x1 ❃ 0✿ Rarefaction waves and vacuum (Einfeldt, Munz, Roe & Sjgreen, 1991): ❧ ✚0 ❃ 0, u0 ❃ 0, E0 ❃ u2

0❂2

❧ Wl = (✚0❀ ✚0u0❀ 0❀ ✚0E0) and Wr = (✚0❀ ✚0u0❀ 0❀ ✚0E0) ❧ If 4✌ 3✌ 1E0 ❃ u2

0, then density and pressure remain positive.

• Y. Penel (LJLL)

Positive schemes for Euler

• 4 / 17

/ / :

• 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)

• Y. Penel (LJLL)

Positive schemes for Euler

• 5 / 17

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• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

MUSCL strategy

Ml Mk Mm Mp Mq Γil Ωi Ωj Mi Mj

nil

Wn+1

i

= Wn

i ∆tn ❳ j✷❱(i)

❥Γij❥ ❥Ωi❥F

• Wn

ij❀ Wn ji❀ nij

✁

• Y. Penel (LJLL)

Positive schemes for Euler

• 6 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Modification of Berthon’s strategy (2006)

Wn+1

i

= Wn

i ∆tn ❳ j✷❱(i)

❥Γij❥ ❥Ωi❥F Wn

ij❀ Wn ji❀ nij

✁

• Y. Penel (LJLL)

Positive schemes for Euler

• 7 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Modification of Berthon’s strategy (2006)

Wn+1

i

= Wn

i ∆tn ❳ j✷❱(i)

❥Γij❥ ❥Ωi❥F Wn

ij❀ Wn ji❀ nij

✁

Wn

i

= ❥Ω✄

i ❥

❥Ωi❥ W✄

i +

❳

j✷❱(i)

❥Ωij❥ ❥Ωi❥ Wn

ij

• Y. Penel (LJLL)

Positive schemes for Euler

• 7 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Modification of Berthon’s strategy (2006)

Wn+1

i

= Wn

i ∆tn ❳ j✷❱(i)

❥Γij❥ ❥Ωi❥F Wn

ij❀ Wn ji❀ nij

✁

Wn

i

= ❥Ω✄

i ❥

❥Ωi❥ W✄

i +

❳

j✷❱(i)

❥Ωij❥ ❥Ωi❥ Wn

ij

Wn+1

i

= ❥Ω✄

i ❥

❥Ωi❥ W

✄ i +

❳

j✷❱(i)

❥Ωij❥ ❥Ωi❥ Wij

• Y. Penel (LJLL)

Positive schemes for Euler

• 7 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Modification of Berthon’s strategy (2006)

Wn+1

i

= Wn

i ∆tn ❳ j✷❱(i)

❥Γij❥ ❥Ωi❥F Wn

ij❀ Wn ji❀ nij

✁

Wn

i

= ❥Ω✄

i ❥

❥Ωi❥ W✄

i +

❳

j✷❱(i)

❥Ωij❥ ❥Ωi❥ Wn

ij

Wn+1

i

= ❥Ω✄

i ❥

❥Ωi❥ W

✄ i +

❳

j✷❱(i)

❥Ωij❥ ❥Ωi❥ Wij W = Wn ∆t ❵

✂

F(Wn❀ Vn❀ n) F(Wn❀ Wn❀ n)✄

for a suitable 1D flux ❋ and a small enough time step ∆t

• Y. Penel (LJLL)

Positive schemes for Euler

• 7 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Modification of Berthon’s strategy (2006)

Wn+1

i

= Wn

i ∆tn ❳ j✷❱(i)

❥Γij❥ ❥Ωi❥F Wn

ij❀ Wn ji❀ nij

✁

Wn

i

= ✑✄

i W✄ i + (1 ✑✄ i )

❳

j✷❱(i)

✑ijWn

ij

Wn+1

i

= ✑✄

i W ✄ i + (1 ✑✄ i )

❳

j✷❱(i)

✑ijWij W = Wn ∆t ❵

✂

F(Wn❀ Vn❀ n) F(Wn❀ Wn❀ n)✄

for a suitable 1D flux ❋ and a small enough time step ∆t

• Y. Penel (LJLL)

Positive schemes for Euler

• 7 / 17

/ / :

• 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 the additional state W✄

i

✷ ❲ Optimization of the CFL condition ”large” ✑✄

i

”small” ✑✄

i

Balance?

• Y. Penel (LJLL)

Positive schemes for Euler

• 7 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Outline

1

Introduction

2

Computation of the time step

3

Admissibility

4

Simulations

• Y. Penel (LJLL)

Positive schemes for Euler

• 8 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Outline

1

Introduction

2

Computation of the time step

3

Admissibility

4

Simulations

• Y. Penel (LJLL)

Positive schemes for Euler

• 9 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Convex combinations

Wn+1

i

= Wn

i

∆tn ❳

j✷❱(i)

❥Γij❥ ❥Ωi❥ F Wn

ij❀ Wn ji❀ nij

✁

Wn

i = ✑✄ i W✄ i + (1 ✑✄ i )

❳

j✷❱(i)

✑ijWn

ij

Wn+1

i

= ✑✄

i W ✄ i + (1 ✑✄ i )

❳

j✷❱(i)

✑ijWij

• Y. Penel (LJLL)

Positive schemes for Euler

• 10 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Convex combinations

Wn+1

i

= Wn

i

∆tn ❳

j✷❱(i)

❥Γij❥ ❥Ωi❥ F Wn

ij❀ Wn ji❀ nij

✁

Wn

i = ✑✄ i W✄ i + (1 ✑✄ i )

❳

j✷❱(i)

✑ijWn

ij

Wn+1

i

= ✑✄

i W ✄ i + (1 ✑✄ i )

❳

j✷❱(i)

✑ijWij Wij = Wn

ij ∆tn 4

❳

k=1

✏ij❀kF Wn

ij❀ Wn ij❀k❀ nij❀k

✁

❀ j ✷ ❱(i)

✽ ❃ ❁ ❃ ✿

Wij =

4

❳

k=1

✏ij❀k ✖ij❀k Wij❀k Wij❀k = Wn

ij ∆tn✖ij❀k

✂

F(Wn

ij❀ Wn ij❀k❀ nij❀k) F(Wn ij❀ Wn ij❀ nij❀k)✄

• Y. Penel (LJLL)

Positive schemes for Euler

• 10 / 17

/ / :

• 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

❥✕k(V ❀ W )❥ ☛0❵. CFL Condition ∆tn ✂ max

j✷❱(i)

✚ ✖✄

ij❀ max 1✔k✔4 ✖ij❀k

✛ ✂ ¯ ✕n

i ✔ ☛0

¯ ✕n

i := max j✷❱(i) 1✔k✔4

✟ ❥un

ij ✁ nij❀k❥ + cn ij ❀ ❥un ij❀k ✁ nij❀k❥ + cn ij❀k

✠ Optimization The optimal coefficient reads ✖♦♣t

i

(✑✄

i ❀ ✑ij) =

✽ ❃ ❃ ❃ ❃ ❁ ❃ ❃ ❃ ❃ ✿ 2 (1 ✑✄

i )❥Ωi❥ max j✷❱(i)

❥Γij❥ ✑ij ❀ si ✑✄

i ✑✄ i

❥❅Ωi❥ ❥Ωi❥ ✔ min

j✷❱(i)

✚ ✑✄

i

✒ 1 2❥Γij❥ ❥❅❚ij❥ ✑ij❥❅Ωi❥ ❥❅❚ij❥ ✓ + ✑ij❥❅Ωi❥ ❥❅❚ij❥ ✛✕1

• Y. Penel (LJLL)

Positive schemes for Euler

• 11 / 17

/ / :

• 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

❥✕k(V ❀ W )❥ ☛0❵. CFL Condition ∆tn ✂ max

j✷❱(i)

✚ ✖✄

ij❀ max 1✔k✔4 ✖ij❀k

✛ ✂ ¯ ✕n

i ✔ ☛0

¯ ✕n

i := max j✷❱(i) 1✔k✔4

✟ ❥un

ij ✁ nij❀k❥ + cn ij ❀ ❥un ij❀k ✁ nij❀k❥ + cn ij❀k

✠ Optimization An optimal bound for the solution is given by: ✖♦♣t

i

✒ ✑✄

i = 1

3❀ ✑ij = ❥Γij❥ ❥❅Ωi❥ ✓ = 3❥❅Ωi❥ ❥Ωi❥ ✿

• Y. Penel (LJLL)

Positive schemes for Euler

• 11 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Example

• Y. Penel (LJLL)

Positive schemes for Euler

• 12 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Outline

1

Introduction

2

Computation of the time step

3

Admissibility

4

Simulations

• Y. Penel (LJLL)

Positive schemes for Euler

• 13 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Key-point

Reconstructed states Wn

i + ∆Wn ij ✷ ❲ provided a suitable ✜-limiter is used.

Making the additional state admissible Wn

ij = Wn i + ∆Wn ij❀

W✄

i = 1

✑✄

i

✷ ✹Wn

i (1 ✑✄ i )

❳

j✷❱(i)

✑ijWn

ij

✸ ✺ = Wn

i 1 ✑✄ i

✑✄

i

❳

j✷❱(i)

✑ij∆Wn

ij

• Y. Penel (LJLL)

Positive schemes for Euler

• 14 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Key-point

Reconstructed states Wn

i + ∆Wn ij ✷ ❲ provided a suitable ✜-limiter is used.

Making the additional state admissible Wn

ij = Wn i + ☞n i ∆Wn ij❀

W✄

i = 1

✑✄

i

✷ ✹Wn

i (1 ✑✄ i )☞n i

❳

j✷❱(i)

✑ijWn

ij

✸ ✺ = Wn

i 1 ✑✄ i

✑✄

i

☞n

i

❳

j✷❱(i)

✑ij∆Wn

ij

☞n

i ✷ [0❀ 1] =

✮ Wn

ij = (1 ☞n i )Wn i + ☞n i

✂ Wn

i + ∆Wn ij

✄ ✷ ❲

• Y. Penel (LJLL)

Positive schemes for Euler

• 14 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Key-point

Reconstructed states Wn

i + ∆Wn ij ✷ ❲ provided a suitable ✜-limiter is used.

Making the additional state admissible Wn

ij = Wn i + ☞n i ∆Wn ij❀

W✄

i = 1

✑✄

i

✷ ✹Wn

i (1 ✑✄ i )☞n i

❳

j✷❱(i)

✑ijWn

ij

✸ ✺ = Wn

i 1 ✑✄ i

✑✄

i

☞n

i

❳

j✷❱(i)

✑ij∆Wn

ij

☞n

i ✷ [0❀ 1] =

✮ Wn

ij = (1 ☞n i )Wn i + ☞n i

✂ Wn

i + ∆Wn ij

✄ ✷ ❲ ✚✄

i ❃ 0❀ p✄ i ❃ 0

= ✮ ☞n

i = min

✚ 1❀ ✑✄

i

1 ✑✄

i

✘✄

i

✛ ✿

• Y. Penel (LJLL)

Positive schemes for Euler

• 14 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Outline

1

Introduction

2

Computation of the time step

3

Admissibility

4

Simulations

• Y. Penel (LJLL)

Positive schemes for Euler

• 15 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

1-2-3 test case

• Y. Penel (LJLL)

Positive schemes for Euler

• 16 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

1-2-3 test case

• Y. Penel (LJLL)

Positive schemes for Euler

• 16 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

1-2-3 test case

• Y. Penel (LJLL)

Positive schemes for Euler

• 16 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

1-2-3 test case

• Y. Penel (LJLL)

Positive schemes for Euler

• 16 / 17

/ / :

• 1. Introduction
• 2. CFL
• 3. Admissibility
• 4. Simulations
• 5. Conclusion

Perspectives

Done Analysis of robustness of general MUSCL strategies Derivation of (sufficient) conditions to preserve positivity Explicit CFL conditions Easy adaptation of industrial codes

• C. Calgaro, E. Creus´

e, T. Goudon & Y. Penel, Positivity-preserving schemes for Euler equations: sharp and practical CFL conditions (under revision). To do ❧ Influence of the numerical flux on ☞n

i

❧ Determination of sets of parameters that activate ☞n

i

❧ Application to other systems of conservation laws

• Y. Penel (LJLL)

Positive schemes for Euler

• 17 / 17

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