Motivation Adaptive semi-Lagrangian schemes Adaptive semi-Lagrangian schemes for transport (how to predict accurate grids ?) Martin Campos Pinto CNRS & University of Strasbourg, France joint work Albert Cohen (Paris 6), Michel Mehrenberger and Eric Sonnendrücker (Strasbourg) MAMCDP Workshop LJLL - Paris, January 2009
Motivation Adaptive semi-Lagrangian schemes Outline Motivation 1 Applications and models for charged particles The Vlasov equation Numerical methods Adaptive semi-Lagrangian schemes 2 Wavelets or mesh refinement ? Dynamic adaptivity The predict-and-readapt scheme
Motivation Adaptive semi-Lagrangian schemes Outline Motivation 1 Applications and models for charged particles The Vlasov equation Numerical methods Adaptive semi-Lagrangian schemes 2 Wavelets or mesh refinement ? Dynamic adaptivity The predict-and-readapt scheme
Motivation Adaptive semi-Lagrangian schemes Introduction Plasma: gas of charged particles (as in stars or lightnings) Applications: controlled fusion, Plane/flame interaction...
Motivation Adaptive semi-Lagrangian schemes Models for plasma simulation F ( t , x , v ) Microscopic model � N body problem in 6D phase space Kinetic models: statistical approach, replace particles { x i ( t ) , v i ( t ) } i ≤ N by a distribution density f ( t , x , v ) binary collisions � Bolztmann equation mean-field approximation � Vlasov equation ∂ t f ( t , x , v ) + v ∂ x f ( t , x , v ) + F ( t , x , v ) ∂ v f ( t , x , v ) = 0 Fluid models: assume f is maxwellian and compute only first moments: density n ( t , x ) := � f d v , momentum f ( v − u ) 2 d v . u ( t , x ) := n − 1 � � vf d v and pressure p :=
Motivation Adaptive semi-Lagrangian schemes Vlasov equation as a "smooth" transport equation Existence of smooth solutions (cf. Iordanskii, Ukai-Okabe, Horst, Wollman, Bardos-Degond, Raviart...) density f is constant along characteristic curves, ( x , v ) ( X , V )( t ; x , v ) Characteristic flow is a measure preserving diffeomorphism F ( t ) : ( x , v ) → ( X , V )( t ; x , v ) B ( t ) : ( X , V )( t ; x , v ) → ( x , v )
Motivation Adaptive semi-Lagrangian schemes Numerical methods for the Vlasov equation { ( x i ( t ) , v i ( t )) : i ≤ N } { f i ( t ) : i ≤ N } Particle-In-Cell (PIC) methods ([Harlow 1955]) Hockney-Eastwood 1988, Birdsall-Langdon 1991 (physics) Neunzert-Wick 1979, Cottet-Raviart 1984, Victory-Allen 1991, Cohen-Perthame 2000 (mathematical analysis) Eulerian (grid-based) methods Forward semi-Lagrangian [Denavit 1972] Backward semi-Lagrangian [Cheng-Knorr 1976, Sonnendrücker-Roche-Bertrand-Ghizzo 1998] Conservative flux based methods [Boris-Book 1976, Fijalkow 1999, Filbet-Sonnendrücker-Bertrand 2001] Energy conserving FD Method: [Filbet-Sonnendrücker 2003]
Motivation Adaptive semi-Lagrangian schemes the Particle-In-Cell method { ( x i ( t ) , v i ( t )) : i ≤ N } Principle: approach the density distribution f by transporting sampled "macro-particles" initialization: deterministic approximation of f 0 � macro-particles { x i ( 0 ) , v i ( 0 ) } i ≤ N knowing the charge and current density, solve the Maxwell system knowing the EM field, transport the macro-particles along characteristics Benefits: intuitive, good for large & high dimensional domains Drawback: sampling in general performed by Monte Carlo � poor accuracy
Motivation Adaptive semi-Lagrangian schemes the (backward) semi-Lagrangian method { f i ( t ) : i ≤ N } Principle: use a transport-interpolation scheme initialization: projection of f 0 on a given FE space knowing f , compute the charge and current densities and solve the Maxwell system Knowing the EM field, transport and interpolate the density along the flow. Benefits: good accuracy, high order interpolations are possible Drawback: needs huge resources in 2 or 3D
Motivation Adaptive semi-Lagrangian schemes Comparison Initializations of a semi-gaussian beam in 1+1 d Solution: adaptive semi-Lagrangian schemes...
Motivation Adaptive semi-Lagrangian schemes Outline Motivation 1 Applications and models for charged particles The Vlasov equation Numerical methods Adaptive semi-Lagrangian schemes 2 Wavelets or mesh refinement ? Dynamic adaptivity The predict-and-readapt scheme
Motivation Adaptive semi-Lagrangian schemes Adaptive semi-Lagrangian scheme (1): adaptive meshes Knowing f n ≈ f ( t n := n ∆ t ), approach the backward flow B ( t n ) : ( x , v ) → ( X , V )( t n ; t n + 1 , x , v ) by a diffeomorphism B n = B [ f n ] transport the numerical solution with T : f n → f n ◦ B n then interpolate on the new mesh M n + 1 : f n + 1 := P M n + 1 T f n M n +1 M n B n ( x, v ) ( x, v ) CP, Mehrenberger, Proceedings of Cemracs 2003
Motivation Adaptive semi-Lagrangian schemes Adaptive semi-Lagrangian scheme (2): wavelets Knowing f n ≈ f ( t n := n ∆ t ), approach the backward flow B ( t n ) : ( x , v ) → ( X , V )( t n ; t n + 1 , x , v ) by a diffeomorphism B n = B [ f n ] transport the numerical solution with T : f n → f n ◦ B n then interpolate on the new grid Λ n + 1 : f n + 1 := P Λ n + 1 T f n Λ n +1 Λ n B n ( x, v ) ( x, v ) Gutnic, Haefele, Paun, Sonnendrücker, Comput. Phys. Comm. 2004
Motivation Adaptive semi-Lagrangian schemes Screenshots
Motivation Adaptive semi-Lagrangian schemes Dynamic adaptivity Problem: Given M n , resp. Λ n , well-adapted to f n , predict M n + 1 , resp. Λ n + 1 , well-adapted to T f n well-adapted: small interpolation error prediction algorithm should be: not too expensive not too long accurate Simple algorithm: predict-and-readapt schemes: predict (a new mesh) transport (the solution) readapt (the mesh) See also Houston, Süli, Technical report 1995, Math. Comp. 2001 Behrens, Iske, and Käser ( ∼ 1996 – 2006)
Motivation Adaptive semi-Lagrangian schemes Prediction by recursive splitting, looking backwards B n M n B n Λ n
Motivation Adaptive semi-Lagrangian schemes Interpolation error estimates hierarchical conforming spaces build on quad meshes and the corresponding P 1 interpolation P M satisfies � ( I − P M ) f � L ∞ � sup | f | W 2 , 1 ( α ) α ∈ M the wavelet interpolation P Λ : f → � γ ∈ Λ d γ ( f ) ϕ γ satisfies � � ( I − P Λ ) f � L ∞ � sup | d γ ( f ) | γ ∈∇ ℓ \ Λ ℓ ≥ 0
Motivation Adaptive semi-Lagrangian schemes ε -adaptivity to f : strong or weak ? weak if the accuracy is meant for the given mesh (or grid) strong if the accuracy is meant for the predicted mesh (or grid)
Motivation Adaptive semi-Lagrangian schemes The predict-and-readapt scheme CP, Mehrenberger, Numer. Math. 2008 CP, Analytical and Num. Aspects of Partial Diff. Eq., de Gruyter, Berlin 2009 given ( M n , f n ) : M n + 1 := T [ B n ] M n ⋄ predict a first mesh ˜ ⋄ perform semi-Lagrangian scheme ˜ f n + 1 := P ˜ M n + 1 T f n ⋄ then re-adapt the mesh M n + 1 := A ε (˜ f n + 1 ) ⋄ and interpolate again f n + 1 := P M n + 1 ˜ f n + 1 Theorem (CP, Mehrenberger) Low-cost prediction algorithms satisfy: M n is ε -adapted to f n T [ B n ] M n is w C ε -adapted to T f n = ⇒ Λ n is ε -adapted to f n T [ B n ]Λ n is w C ε -adapted to T f n = ⇒ Some papers available at http://www-irma.u-strasbg.fr/ campos
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