Some nice features of AP-schemes Anisotropic transport equations - PowerPoint PPT Presentation

1 Some nice features of AP-schemes Anisotropic transport equations Claudia Negulescu Institut de Mathématiques de Toulouse Université Paul Sabatier C. Negulescu, 22/05/2018

Introduction/Motivation 2 Objective: Numerical study of highly anisotropic, multi-scale problems • Many pb. in nature exhibit multi-scale behaviours, which can be rather different in character • Typical: occurence of one or several small/large parameters (Reynolds, Peclet, Mach nbr. etc) • General, unified treatment is impossible C. Negulescu, 22/05/2018

Multi-scale plasma dynamics 3 Plasma dynamics is characterized by multi-scale phenomena ⇒ Strong magn. fields create anisotropies ⇒ Particles gyrate around the field lines Hybrid models Kinetic models Fluid models τpe τce τpi τci τa τcs τei τ a : Alfen wave period τ pe,pi : Inv. electr./ion plasma freq. τ cs : Ion sound period τ ce,ci : Electr./ion cyclotron period τ ei : Electr-ion collision time λ D : Debye length ρ e,i : Electr./ion Larmor radius ω pe,pi : Electr./ion plasma frequency δ e,i = c/ω pe,pi : Electr./ion skin depth c : sound speed λD ρe δe ρi δi L C. Negulescu, 22/05/2018

Multi-scale problems 4 A small-scale numerical simulation is out of reach ➠ requires mesh-sizes dependent on small scale param. ε ≪ 1 ➠ excessive computational time and memory space are needed to capture small scales It is not always of interest to resolve the details at the small scale. Multi-scale strategies are much more adequate! ➠ homogeneisation, domain decomposition, multi-grids, multi-scale methods based on wavelets or finite elements, multi-scale variational methods Essential feature of these methods ➠ capture efficiently the large scale behavior of the solution, without resolving the small scale features C. Negulescu, 22/05/2018

Asymptotic Preserving schemes 5 Difficulty: Resolution of multiscale pb. can be very difficult, if the pb. becomes singular, as one of the parameters ε → 0 ➠ ( P ε ) sing. perturbed pb. with sol. f ε ; ➠ the seq. f ε converges towards f 0 , sol. of a limit pb. ( P 0 ) ; ➠ the limit pb. ( P 0 ) is different in type from the initial ( P ε ) ; ➠ standard schemes would require ∆ t, ∆ x ∼ ε for stability. Definition: A scheme P ε,h is AP iff it is convergent for h → 0 uniformely in ε , i.e. h → 0 P ε P ε,h ε → 0 ε → 0 h → 0 P 0 ,h P 0 C. Negulescu, 22/05/2018

Asymptotic Preserving schemes 6 AP-procedure: ➠ requires that the limit problem ( P 0 ) is identified and well-posed; ➠ consists in trying to mimic at discrete level the asymptotic behaviour of the sing. perturbed pb. sol. f ε ; ➠ requires a sufficient degree of implicitness (not obvious). Advantages: ➠ gives accurate and stable results, with no restrictions on the computational mesh; ➠ enables to capture automatically the Limit model P 0 , if ε → 0 (micro-macro transition); ➠ no more coupling needed, if ε ( x ) is variable. C. Negulescu, 22/05/2018

Kinetic models and specific limit regimes 7 Fundamental kinetic model: Vlasov/Boltzmann equation ∂ t f + v · ∇ x f + q m ( E + v × B ) · ∇ v f = Q ( f ) Several small scales/parameters occur, leading to diff. regimes: • Hydrodynamic scaling [Filbet/Jin; Dimarco/Pareschi] ∂ t f + v · ∇ x f = 1 εQ ( f ) ➠ 0 < ε ≪ 1 : mean free path (Knudsen nbr.) ➠ in the limit ε → 0 , one gets the compressible Euler eq. ➠ AP-scheme: Decomposition of the source term in stiff- and non-stiff part Q ( f ) = Q ( f ) − P ( f ) + P ( f ) ε ε ε C. Negulescu, 22/05/2018

Kinetic models and specific limit regimes 8 • Drift-Diffusion scaling [Klar; Lemou/Mieussens] ∂ t f + 1 ε ( v · ∇ x f + E · ∇ v f ) = 1 ε 2 Q ( f ) ➠ 0 < ε ≪ 1 : mean free path; long-time asymp. ➠ in the limit ε → 0 , one gets the Drift-Diffusion model ➠ AP-scheme: Micro-Macro decomp. f = ρM + εg • High-field limit, strong magn. fields [Bostan, Frenod, Golse, Saint-Raymond] ∂ t f + v · ∇ x f + E · ∇ v f + 1 ε ( v × B ) · ∇ v f = 0 ➠ 0 < ε ≪ 1 : cyclotronic period; strong B -field; ➠ in the limit ε → 0 , one gets the gyro-kinetic model. C. Negulescu, 22/05/2018

Kinetic models and specific limit regimes 9 • Adiabatic scaling [Negulescu,...] ∂ t f + 1 εv · ∇ x f − 1 ε ( E + 1 εv × B ) · ∇ v f = 1 εQ ( f ) ➠ 0 < ε ≪ 1 : small electron/ion mass ratio, collisionality, strong B-fields ➠ in the limit ε → 0 , one gets the electr. Boltzmann rel. Diff. regimes, Diff. kind of asymptotic behaviour as ε → 0 : ➠ diffusive behaviour (HD,DD) ➠ highly oscillating behaviour (BE) ⇒ different kinds of num. schemes required ! C. Negulescu, 22/05/2018

Evolution pb. with stiff transport terms 10 • Aim: Efficient num. resolution of multi-scale pb. of the type: ∂ t f ε + b ε · ∇ f ε + L f ε = 0 • Motivating physical models: ➠ Anisotropic Fokker-Planck eq. in the gyro-kinetic scaling: i + 1 ∂ t f ε i + v · ∇ x f ε i + E · ∇ v f ε ε ( v × B ) · ∇ v f ε i = η ∇ v · [ v f ε i + ∇ v f ε i ] . ➠ Vlasov-Poisson eq. in the long-time asymptotics: ∂ t f e + 1 ε v ∂ x f e − 1  ∀ t ∈ R + , ∀ ( x, v ) ∈ Ω ⊂ R 2 ε E ( t, x ) ∂ v f e = 0 ,    �  − ∂ xx ϕ = 1 − n e , n e ( t, x ) = f e ( t, x, v ) dv , E = − ∂ x ϕ .   R ➠ Euler 2D eq. / Vorticity eq. in the long-time asymptotics: ∂ t ω ε + 1 ε u ε · ∇ ω ε = 0 , − ∆Ψ ε = ω ε , u ε = ⊥ ∇ Ψ ε . C. Negulescu, 22/05/2018

11 I. Vlasov eq. in the gyro-kinetic regime Work based on: [1] B. Fedele, C. Negulescu, Numerical study of an anisotropic Vlasov equation arising in plasma physics , to appear in KRM (Kinetic and Related Models), 2018. C. Negulescu, 22/05/2018

Introduction/Motivation 12 Starting model: Anisotropic Vlasov eq. (gyro-kinetic regime) ∂ t f ε + v · ∇ x f ε + E · ∇ v f ε + 1 ε ( v × B ) · ∇ v f ε = 0 Aim: design efficient numerical scheme ➠ accuracy and stability independent on ε (AP property); ➠ rapid, not time and memory consuming simulations; ➠ simple implementation, practical scheme. Important questions: ➠ what is the asymptotic behaviour of the solution f ε as ε → 0 ? ➠ what does one want to see in the asymptotic limit? All microscopic information or only the macroscopic information? C. Negulescu, 22/05/2018

Asymptotic behaviour as ε → 0 13 B ( x ) • Magnetic field B : direction b ( x ) := | B ( x ) | ; magnitude � ( x ) := | B ( x ) | ; ∇ · B = 0 . • Dominant operator: T := ( v × B ) · ∇ v T : D ( T ) → L 2 (Ω × R 3 ) , f ∈ L 2 (Ω × R 3 ) / T f ∈ L 2 (Ω × R 3 ) � � D ( T ) := . • Characteristics: C x , v := { ( X ( s ; x , v ) , V ( s ; x , v )) , s ∈ R } dX  ds = 0 ,    dV  ds = � ( X ( s )) V ( s ) × b ( X ( s )) ,   V ( s ; x , v ) = cos( � ( x ) s ) v ⊥ + sin( � ( x ) s ) ⊥ v + v || , X ( s ; x , v ) = x , ∀ s ∈ R 2 π ➠ periodic trajectories with period T c ( x ) := � ( x ) ; ➠ invariants: x , | v ⊥ | and v || ; ➠ ker T := { f ∈ L 2 (Ω × R 3 ) / ∃ g : Ω × R × R + → R st. f ( x , v ) = g ( x , v b , | v ⊥ | ) } . C. Negulescu, 22/05/2018

Asymptotic behaviour as ε → 0 14 b := { ̟ ∈ R 3 / | ̟ | = 1 , ̟ · b = 0 } S 1 • Cylindrical coordinates with respect to b : ̟ := v ⊥ | v ⊥ | ∈ S 1 v = v || + v ⊥ = v b b + r ̟ , r := | v ⊥ | , b . J : L 2 (Ω × R 3 ) → ker( T ) • Gyro-average operator: (orthog. proj. on ker( T ) ) � T c ( x ) 1 J ( f )( x , v ) := f ( X ( s ; x , v ) , V ( s ; x , v )) ds T c ( x ) 0 1 � = f ( x , v b b + | v ⊥ | ̟ ) d̟ . 2 π S 1 b L 2 (Ω × R 3 ) = ker( T ) ⊕ ⊥ ker( J ) f = J ( f ) + f ′ • Decomposition : T : D ( T ) ∩ ker( J ) → ker( J ) , bij. map • Limit model for ε → 0 :  f 0 ∈ ker( T ) ( v × B ) · ∇ v f 0 = 0 i.e.  ∂ t f 0 + J ( v · ∇ x f 0 ) + J ( E · ∇ v f 0 ) = 0 .  C. Negulescu, 22/05/2018

Simple toy model 15 • Simplified toy model: ∂ t f ε + a ∂ x f ε + b  ε ∂ y f ε = 0 , ∀ ( t, x, y ) ∈ [0 , T ] × [0 , L x ] × [0 , L y ] ,   ( V ) ε  f ε (0 , x, y ) = f in ( x, y ) = sin( x ) � � cos(2 y ) + 1 .  • Exact solution: � � ex ( t, x, y ) = f in ( x − at, y − b y − b � � �� f ε � � ε t ) = sin x − at cos 2 εt + 1 , 2 ε = 1 (exact) ε = 0.5 ε = 0.1 1.5 1 ex (t,x Nx−1 ,y Ny−1 ) 0.5 0 −0.5 f ε −1 −1.5 −2 0 2 4 6 8 10 12 t C. Negulescu, 22/05/2018

Simple toy model 16 • Limit model:  ∂ t f 0 + a∂ x f 0 = 0 , ∀ ( t, x ) ∈ [0 , T ] × [0 , L x ] ,   f 0 (0 , x ) = ¯ f in ( x ) , ∀ x ∈ [0 , L x ] ,   � L y 1 f ε ( t, x ) := ¯ f ε ( t, x, y ) dy . Average or Projection: L y 0 • Exact limit solution: f 0 ( t, x ) = ¯ � � f in ( x − at ) = sin x − at . 1 0.8 0.6 0.4 0.2 0 4 2 3 2 0 1 0 -2 -1 -2 -4 -3 C. Negulescu, 22/05/2018

17 Numerical schemes ➠ Field aligned or NOT-field aligned configuration ➠ Micro-Macro scheme (Fourier schemes) ➠ IMEX or Implicit schemes ➠ Lagrange-Multiplier scheme (NEW SCHEME!) C. Negulescu, 22/05/2018