HYP 2012 A positive, well-balanced and entropy-satisfying scheme - PowerPoint PPT Presentation

HYP 2012 A positive, well-balanced and entropy-satisfying scheme for shallow water flows Interest of the kinetic description E. Audusse, M.-O. Bristeau, C. Pares & J. Sainte-Marie Padova - june 2012

Introduction Kinetic description & scheme Numerical validations Outline & Main ideas Introduction Kinetic description & num. scheme general scheme with discrete entropy Numerical validations * * * * * * * * * * • Interest of efficient numerical methods ◦ in fluid mechanics, geophysics ◦ non smooth solutions, few dissipation • Useful in practice (simple): for scientists, industrial

Introduction Kinetic description & scheme Numerical validations Seism : Japan, march 2011 source IPGP (A. Mangeney)

Introduction Kinetic description & scheme Numerical validations Comparison with DART buoys (3d hyd. Navier-Stokes) Long distance small amplitude ⇒ accurate scheme is needed

Introduction Kinetic description & scheme Numerical validations Japan tsunami simulated with Saint-Venant • Hydrostatic reconstruction vs. proposed scheme ◦ Unstructured mesh, 2.10 6 nodes, 1 st order scheme (space & time)

Introduction Kinetic description & scheme Numerical validations The Saint-Venant system � ∂ t + ∂ ( H ¯ u ) ∂ H = 0 ∂ x ( SV ) � 2 H 2 � u 2 + g ∂ ( H ¯ u ) + ∂ = − gH ∂ z b H ¯ ∂ t ∂ x ∂ x • The system is hyperbolic • The water depth satisfies � d H ≥ 0 , H = 0 dt • Static equilibrium, “lake at rest” u = 0 , H + z b = Cst • It admits a convex entropy (the energy) � � � � u 2 u 2 ∂ H ¯ 2 + g + ∂ H ¯ 2 H 2 + gHz b 2 + gH 2 + gHz b ∂ x u ≤ 0 ∂ t ⇒ Positivity, well-balancing, consistency, discrete entropy . . . without reconstruction

Introduction Kinetic description & scheme Numerical validations Num. methods for the Saint-Venant system • Finite volume schemes [Bouchut’04] • Various solvers (relaxation, Roe, HLL, kinetic,. . . ) • Well-balanced scheme required ∂ H ∂ t + ∂ ( H ¯ u ) i − ∆ t ⇒ H n + 1 = H n ∆ x ( F n i + 1 / 2 − F n = 0 , i − 1 / 2 ) i ∂ x max ( √ gH i , √ gH i + 1 ) with e.g. F n i + 1 / 2 = ( H i − H i + 1 ) � = 0 2 when H j + z b , j = Cst • Hydrostatic reconstruction [ABBKP,04] ◦ z ∗ b , j , z ∗ b , j + 1 ⇒ H ∗ j = H ∗ j + 1 at rest ◦ efficient, various situations ◦ only semi discrete entropy

Introduction Kinetic description & scheme Numerical validations Kinetic representation of the Saint-Venant system � � � ξ − ¯ u • Gibbs equilibrium M ( x , t , ξ ) = H c χ with c = gH / 2 c � � R ω 2 χ ( ω ) = 1 where χ ( ω ) = χ ( − ω ) ≥ 0, supp ( χ ) ⊂ Ω , R χ ( ω ) = Proposition (Audusse, Bristeau, Perthame 04) The functions ( H , ¯ u , E )( t , x ) are strong solutions of the Saint-Venant system if and only if M ( x , t , ξ ) is solution of the kinetic equation ∂ M ∂ t + ξ ∂ M ∂ x − g ∂ z b ∂ M ( B ) , ∂ξ = Q ( x , t , ξ ) ∂ x where Q ( t , x , ξ ) is a collision term. � • Macroscopic variables ( H , ¯ R ( 1 , ξ, ξ 2 / 2 ) M d ξ u , E ) = • A linear transport equation . . . easy to upwind

Introduction Kinetic description & scheme Numerical validations Discrete scheme for the Saint-Venant system (I) � � i = H n u n ξ − ¯ • Gibbs equilibrium M n i χ i i c n c n i • A simple upwind scheme, for a given ξ � ∂ M n i + 1 / 2 M n + 1 − M n i − σ n ξ ( M n i + 1 − M n = i ) ✶ ξ ≤ 0 − g ∆ z b , i + 1 / 2 i i ∂ξ � ∂ M n i − 1 / 2 + ξ ( M n i − M n i − 1 ) ✶ ξ ≥ 0 − g ∆ z b , i − 1 / 2 ∂ξ with M n i + 1 / 2 = M n i + 1 / 2 − ✶ ξ ≤ 0 + M n i + 1 / 2 + ✶ ξ ≥ 0 M n i + 1 / 2 − , M n to be defined later i + 1 / 2 − • Key point � � � ∂ M i + 1 / 2 + ∂ M i + 1 / 2 − ∂ M ∂ξ d ξ = 0 , but d ξ + d ξ � = 0 ∂ξ ∂ξ ❘ ξ ≤ 0 ξ ≥ 0

Introduction Kinetic description & scheme Numerical validations Discrete scheme for the Saint-Venant system (II) • Extended version of an idea in Perthame-Simeoni’01 � � � ∂ξ − ξ ∂ � g ∂ z b ∂ M M ξ p d ξ = 0 , p = 0 , 1 , ∂ x ∂ x ❘ � � � with � ξ � H M = c χ , H = η − z b , η = Cst � � c • The proposed scheme is � � M n + 1 − M n i − σ n M n i + 1 / 2 − M n = i i i − 1 / 2 with i + 1 / 2 − ξ � M n ξ M n M n = i + 1 / 2 i + 1 / 2 M n M n i ✶ ξ ≥ 0 + M n = i + 1 ✶ ξ ≤ 0 i + 1 / 2 � � i + 1 / 2 + ✶ ξ ≤ 0 + � M n M n M n = i + 1 / 2 − ✶ ξ ≥ 0 i + 1 / 2 � � � H n ξ i + 1 / 2 − � � M n H n = χ , i + 1 / 2 − = η i + 1 / 2 − z b , i i + 1 / 2 − c n c n � � i + 1 / 2 − i + 1 / 2 −

Introduction Kinetic description & scheme Numerical validations Discrete scheme for the Saint-Venant system (III) • The proposed scheme is � � M n + 1 − M n i − σ n M n i + 1 / 2 − M n = i i − 1 / 2 i with i + 1 / 2 − ξ � M n ξ M n M n = i + 1 / 2 i + 1 / 2 M n M n i ✶ ξ ≥ 0 + M n = i + 1 ✶ ξ ≤ 0 i + 1 / 2 � � i + 1 / 2 + ✶ ξ ≤ 0 + � M n M n M n = i + 1 / 2 − ✶ ξ ≥ 0 i + 1 / 2 � � � H n ξ i + 1 / 2 − � � M n H n = χ , i + 1 / 2 − = η i + 1 / 2 − z b , i i + 1 / 2 − c n c n � � i + 1 / 2 − i + 1 / 2 − • Macroscopic scheme � � H n + 1 ❘ M n + 1 − H n + 1 u n + 1 ❘ ξ M n + 1 − = d ξ, = d ξ i i i i i • only analytic quadrature formula

Introduction Kinetic description & scheme Numerical validations Properties of the scheme • Key point � � � ∂ M i + 1 / 2 + ∂ M i + 1 / 2 − ∂ M ∂ξ d ξ = 0 , but d ξ + d ξ � = 0 ∂ξ ∂ξ ❘ ξ ≤ 0 ξ ≥ 0 • Well-balanced ◦ trivial • Positive ◦ the CFL does not depend on ∂ z b ∂ x ◦ well behaves when H → 0 • Consistency • 2 nd order in time (Modified Heun) and space (centered term) • Convergence rate : C ∆ x vs. c ∆ x with c < C c n • With modified ˆ i + 1 / 2 ± : can be used with other FV solvers (HLL, Rusanov) • No discrete entropy

Introduction Kinetic description & scheme Numerical validations Scheme for H ≥ | ∆ z b | (discrete entropy) - I • Gibbs equilibrium � � � � ξ − ¯ u ξ − ¯ u ◦ M ( x , t , ξ ) = H M ( x , t , ξ ) = H c χ 0 , c φ χ 0 c c � � + ∞ 1 − z 2 ◦ χ 0 ( z ) = 1 4 , φ χ 0 ( z ) = z 1 χ 0 ( z 1 ) dz 1 π z • χ 0 is the minimum of the set (energy), see [Perthame-Simeoni 01] � ξ 2 � � 2 f ( ξ ) + g 2 8 f 3 ( ξ ) + gz b f ( ξ ) E ( f ) = d ξ ❘ • Modified Boltzmann equation ∂ M ∂ t + ξ ∂ M ∂ x − g ∂ z b ∂ M ∂ξ = Q ∂ x ∂ M ∂ t + ξ ∂ M ∂ x + g ξ − u M ∂ z b ⇔ ∂ x = Q c 2 ∂ ξ eliminated in the Boltzmann equation...

Introduction Kinetic description & scheme Numerical validations Scheme for H ≥ H 0 > 0 (discrete entropy) - II • Goal : M n + 1 − as a convex combination of M n i − 1 , M n i and M n i + 1 i • A simple upwind scheme, for a given ξ � � M n + 1 − M n i − σ n M n i + 1 / 2 − M n = i i i − 1 / 2 with M n M n i + 1 / 2 + + M n = i + 1 / 2 i + 1 / 2 − � � ξ − u n ξ + 2 ∆ z b , i + 1 / 2 i + 1 M n c n ✶ ξ ≤ ξ i + 1 / 2 + M n = i + 1 / 2 + i + 1 / 2 + i + 1 H n c n i + 1 i + 1 � � ξ + 2 ∆ z b , i + 1 / 2 ξ − u n M n i c n ✶ ξ ≥ ξ i + 1 / 2 − M n = i + 1 / 2 − i + 1 / 2 − i H n c n i i • So M n + 1 − = ( 1 − A n i ) M n i + A n i − 1 / 2 + M n i − 1 + A n i + 1 / 2 − M n i + 1 i with A n j ≥ 0, 1 − A n i ≥ 0

Introduction Kinetic description & scheme Numerical validations Scheme for H ≥ H 0 > 0 (discrete entropy) - III • The scheme is well-balanced, consistent and positive Proposition Let us consider a real convex function e ( . ) defined over ❘ + . Under the CFL condition, the scheme satisfies the in-cell entropy inequality � � E n + 1 ≤ E n Λ n i + 1 / 2 − Λ n i + σ i i − 1 / 2 i � with E n e ( M n = i ) d ξ i ❘ � � � Λ n σ n A n i + 1 / 2 − e ( M n i + 1 ) − A n i e ( M n = i ) d ξ i + 1 / 2 i ❘ 8 f 3 + gz b f gives a discrete version In particular the choice e ( f ) = ξ 2 2 f + g 2 of the energy balance.

Introduction Kinetic description & scheme Numerical validations Numerical validations • Only analytical solutions • Stationary/transient, continuous/discontinuous solutions • 1 st and 2 nd order schemes ⇒ not exhaustive validations • Two main ideas ◦ Systematic biais & accuracy ◦ fluvial regime over a bump (anim) ◦ general scheme

Introduction Kinetic description & scheme Numerical validations Other solvers • HLL, kinetic & Rusanov fluxes • 1 st and 2 nd order schemes • fluvial regime over a bump • general scheme

Introduction Kinetic description & scheme Numerical validations Transcritical regime with shock • HLL & kinetic fluxes • general scheme

Introduction Kinetic description & scheme Numerical validations Parabolic bowl • Kinetic fluxes • 1 st and 2 nd order (in space & time) schemes • general scheme (anim)