On the stability of IMEX schemes for singular hyperbolic PDEs - PowerPoint PPT Presentation

On the stability of IMEX schemes for singular hyperbolic PDE’s Sebastian Noelle, RWTH Aachen joint with Klaus Kaiser, Ruth Sch¨ obel, Jochen Sch¨ utz, Hamed Zakerzadeh Paris, Nov. 2015 Sebastian Noelle AP Stability Paris, Nov. 2015 1 / 25

Key example Isentropic gas dynamics ∂ t ρ + div ( ρ u ) = 0 , ∂ t ( ρ u ) + div ( ρ u ⊗ u ) + 1 ε 2 ∇ p ( ρ ) = 0 . Sebastian Noelle AP Stability Paris, Nov. 2015 2 / 25

Key example Isentropic gas dynamics ∂ t ρ + div ( ρ u ) = 0 , ∂ t ( ρ u ) + div ( ρ u ⊗ u ) + 1 ε 2 ∇ p ( ρ ) = 0 . Mach number: ε = u ref c ref Sebastian Noelle AP Stability Paris, Nov. 2015 2 / 25

Challenges Sebastian Noelle AP Stability Paris, Nov. 2015 3 / 25

Challenges ε ≪ 1: stiffness u ⋅ n , u ⋅ n ± c ε Sebastian Noelle AP Stability Paris, Nov. 2015 3 / 25

Challenges ε ≪ 1: stiffness u ⋅ n , u ⋅ n ± c ε ε → 0: change of type compressible to incompressible flow Klainerman-Majda Sebastian Noelle AP Stability Paris, Nov. 2015 3 / 25

Challenges Sebastian Noelle AP Stability Paris, Nov. 2015 4 / 25

Challenges Preserve the Asymptotics Asymptotic Consistency Asymptotic Stability ⇒ AP property (Shi Jin) Sebastian Noelle AP Stability Paris, Nov. 2015 4 / 25

Challenges Preserve the Asymptotics Asymptotic Consistency Asymptotic Stability ⇒ AP property (Shi Jin) Efficiency: implicit for stiff part explicit for non-stiff part ⇒ IMEX Sebastian Noelle AP Stability Paris, Nov. 2015 4 / 25

Todays Talk Sebastian Noelle AP Stability Paris, Nov. 2015 5 / 25

Todays Talk ● a key stability structure Sebastian Noelle AP Stability Paris, Nov. 2015 5 / 25

Todays Talk ● a key stability structure ● a new class of IMEX schemes Sebastian Noelle AP Stability Paris, Nov. 2015 5 / 25

Todays Talk ● a key stability structure ● a new class of IMEX schemes ● examples, applications, stability Sebastian Noelle AP Stability Paris, Nov. 2015 5 / 25

Linear Stability Theory Prototype linear system U t + AU x = 0 . with stiff eigenvalues λ max ∶= max ∣ λ ∣ = O ( 1 ε ) λ min ∶= min ∣ λ ∣ = O ( 1 ) Sebastian Noelle AP Stability Paris, Nov. 2015 6 / 25

Linear Stability Theory Admissible Splittings Definition A splitting A = ̃ A + ̂ A . is admissible , if (i) both ̃ A and ̂ A induce a hyperbolic system (ii) ̃ λ ∶ = ρ (̃ A ) = O ( 1 ε ) ̂ λ ∶ = ρ (̂ A ) = O ( 1 ) Sebastian Noelle AP Stability Paris, Nov. 2015 7 / 25

Linear Stability Theory CFL Conditions ν ∶= λ max ∆ t full CFLnumber ∆ x ν ∶= ̂ ̂ λ ∆ t nonstiff CFLnumber ∆ x ν = O ( 1 ) ⇒ ̂ ν = O ( ε ) stable inefficient ν = O ( 1 ε ) ⇐ ̂ ν = O ( 1 ) unstable efficient Sebastian Noelle AP Stability Paris, Nov. 2015 8 / 25

Linear Stability Theory Flux-Splitting & IMEX Time-Discretization Implicit-explicit discretization Klein 1996 Degond, Tang 2011 Haack, Jin, Liu 2011 U n + 1 = U n + ̃ + ̂ AU n + 1 AU n x x Sebastian Noelle AP Stability Paris, Nov. 2015 9 / 25

Linear Stability Theory Examples of stability/instability Numerical experiments: IMEX schemes which are - based on admissible splittings - asymptotic consistent can be stable and unstable Noelle, Bispen, Arun, Lukacova, Munz SISC 2014 Euler, Low Mach, IMEX, weakly AP Bispen, Arun, Lukacova, Noelle CiCP 2014 Shallow water, Low Froude, IMEX, AP Sebastian Noelle AP Stability Paris, Nov. 2015 10 / 25

Linear Stability Theory Modified equation, cf. Warming/Hyett 1974 Theorem (Sch¨ utz, Noelle JSC 2014) The modified equation of the IMEX scheme is w t + Aw x = ∆ t 2 C w xx with diffusion matrix ∆ t I − (̂ A − ̃ A )(̂ A + ̃ C ∶= (̂ α + ̃ α ) ∆ x A ) and numerical upwind viscosities ̂ α , ̃ α . Sebastian Noelle AP Stability Paris, Nov. 2015 11 / 25

Linear Stability Theory the crucial commutator Is C positive definite? ∆ t I − ̂ (̃ A ̂ A − ̂ A ̃ ̃ = ((̂ α + ̃ α ) ∆ x A 2 ) + A ) + A 2 C = O ( 1 ) + O ( 1 ε ) + O ( 1 ε 2 ) A ] = 0 Certainly yes, if commutator [̃ A , ̂ Sebastian Noelle AP Stability Paris, Nov. 2015 12 / 25

Linear Stability Theory Example (Sch¨ utz, Noelle 2014) Fourier stability analysis for prototype system ⎛ ⎞ a 1 0 A = ⎜ ⎟ 1 1 a ⎝ ⎠ ε 2 ε 2 0 1 a a > 0, eigenvalues √ λ = a , a ± 2 ε Sebastian Noelle AP Stability Paris, Nov. 2015 13 / 25

Linear Stability Theory Euler: classical versus characteristic splitting Allowable timestep sizes - A comparison 10 5 10 2 ν 10 − 1 10 − 4 Splitting by Arun, Noelle... Characteristic splitting 10 − 7 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 ε Comparison of classical versus characteristic splitting Sebastian Noelle AP Stability Paris, Nov. 2015 14 / 25

Linear Stability Theory How to recover stability Need e.g. ̂ A and ̃ A symmetric or A ̂ ̃ A − ̂ A ̃ A = O ( 1 ) or ̂ A = O ( ε ) Sebastian Noelle AP Stability Paris, Nov. 2015 15 / 25

Linear Stability Theory Theorem (Stability for Haack-Jin-Liu (Zakerzadeh 2015)) For the isentropic Euler equations, the Haack-Jin-Liu scheme with Mach-uniform CFL condition, has (strictly) stable modified equation in the sense of Majda-Pego, i.e. it is AP stable. Sebastian Noelle AP Stability Paris, Nov. 2015 16 / 25

RS-IMEX Reference-Solution IMEX Nonlinear hyperbolic system of balance laws ∂ t U ( x , t ; ε ) + ∇ ⋅ F ( U , x , t ; ε ) = S ( U , x , t ; ε ) with U ∶ R d × R + × ( 0 , 1 ] → R m , ( x , t ; ε ) ↦ U ( x , t ; ε ) ● Challenge: Stiffness as ε → 0 ● Goal: Asymptotic stability Sebastian Noelle AP Stability Paris, Nov. 2015 17 / 25

RS-IMEX Reference solution and scaled perturbation: U = U + D V R d × R + U ∶ ( x , t ) U ( x , t ) → R m , R d × R + × ( 0 , 1 ] ↦ V ∶ ( x , t ; ε ) U ( x , t ; ε ) → R m , ↦ and D = diag ( ε k 1 ,...,ε k m ) Taylor expansion with remainder of F and S around U : F = F ( U ) + A ( U ) DV + ̂ F ( U , V ) = D ( G + ̃ G + ̂ G ) ′ V DV + ̂ S ( U , V ) = D ( Z + ̃ Z + ̂ Z ) S = S ( U ) + S �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� RS + IM + EX Sebastian Noelle AP Stability Paris, Nov. 2015 18 / 25

RS-IMEX Theorem (Modified equation for RS-IMEX (Noelle 2014)) B 0 W t = −∇ ⋅ B 1 + B 2 + ∇ ⋅ ( B 3 ⋅ ∇ W ) with Z ′ − ̂ 2 (̃ B 0 ∶ = I − ∆ t Z ′ ) , G ′ − ̂ Z ′ + ̂ Z ′ − ̃ B 1 ∶ = ̃ G + ̂ 2 ((̃ G ′ )(̃ G x − ̂ G + ∆ t G x )) , B 2 ∶ = ̃ Z + ̂ 2 (̃ Z t − ̂ Z + ∆ t Z t ) , B 3 ∶ = (̂ α + ¯ α ) ∆ x G ′ − ̂ G ′ + ̂ 2 (̃ G ′ )(̃ I + ∆ t G ′ ) . 2 Study this for each application! Sebastian Noelle AP Stability Paris, Nov. 2015 19 / 25

RS-IMEX RS-IMEX is AP for isentropic Euler Theorem (Consistency for RS-IMEX (Zakerzadeh 2015)) For isentropic Euler equations, the RS-IMEX scheme is consistent with the asymptotic limit in the fully-discrete settings, i.e. it is AP consistent. (Sch¨ obel 2015) AP consistency for semi-discrete scheme Theorem (Stability for RS-IMEX (Zakerzadeh 2015)) For isentropic Euler equations, the RS-IMEX scheme with Mach-uniform CFL condition, has (strictly) stable modified equation in the sense of Majda-Pego, i.e. it is AP stable. Sebastian Noelle AP Stability Paris, Nov. 2015 20 / 25

RS-IMEX van der Pol and IMEX (Sch¨ utz, Kaiser 2015) Prototype example 5 ( y ′ z ′ ) = ( ) . z g ( y , z ) 0 ε ε = 1 − 5 ε = 0 . 5 ’Traditional’ splitting: ( ) + ( z 0 ) ε = 0 . 3 0 − 2 − 1 0 1 2 g ( y , z ) ε Sebastian Noelle AP Stability Paris, Nov. 2015 21 / 25

RS-IMEX van der Pol and IMEX ’Reference solution’ ( RS ) ε → 0: ( y ′ 0 ) = ( g ( y ( 0 ) , z ( 0 ) )) . z ( 0 ) ( 0 ) RS-IMEX splitting based on w ( 0 ) : f ( w ) = f ( w ( 0 ) ) + f ′ ( w ( 0 ) )( w − w ( 0 ) ) + Rest Motivation: w − w ( 0 ) = O ( ε ) . Sebastian Noelle AP Stability Paris, Nov. 2015 22 / 25

RS-IMEX RS-IMEX + Runge-Kutta 10 − 1 10 − 1 ε = 10 − 1 ε = 10 − 1 10 − 2 ε = 10 − 3 ε = 10 − 3 10 − 3 10 − 3 ε = 10 − 5 ε = 10 − 5 ε = 10 − 7 ε = 10 − 7 10 − 5 10 − 5 10 − 4 Error Error Error 10 − 7 10 − 7 10 − 6 ε = 10 − 1 10 − 9 10 − 9 ε = 10 − 3 ε = 10 − 5 10 − 8 ε = 10 − 7 10 − 11 10 − 11 10 − 3 10 − 2 10 − 1 10 − 3 10 − 2 10 − 1 10 − 3 10 − 2 10 − 1 Size of ∆ t Size of ∆ t Size of ∆ t 10 − 1 10 − 1 ε = 10 − 1 ε = 10 − 1 10 − 2 ε = 10 − 3 ε = 10 − 3 10 − 3 10 − 3 ε = 10 − 5 ε = 10 − 5 ε = 10 − 7 ε = 10 − 7 10 − 4 10 − 5 10 − 5 Error Error Error 10 − 7 10 − 7 10 − 6 ε = 10 − 1 10 − 9 10 − 9 ε = 10 − 3 ε = 10 − 5 10 − 8 ε = 10 − 7 10 − 11 10 − 11 10 − 3 10 − 2 10 − 1 10 − 3 10 − 2 10 − 1 10 − 3 10 − 2 10 − 1 Size of ∆ t Size of ∆ t Size of ∆ t (Left to right) DPA-242, BHR-553, BPR-353. (Top to bottom) Standard / RS-IMEX IMEX Runge-Kutta (Pareschi, Russo, Boscarino ...) standard splitting looses convergence order RS-IMEX gives full order of accuracy Sebastian Noelle AP Stability Paris, Nov. 2015 23 / 25