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
Outline 1 Applications and models for charged particles The Vlasov equation Numerical methods 2 Wavelets or mesh refinement? Dynamic adaptivity The predict-and-readapt scheme
Outline 1 Applications and models for charged particles The Vlasov equation Numerical methods 2 Wavelets or mesh refinement? Dynamic adaptivity The predict-and-readapt scheme
Introduction Plasma: gas of charged particles (as in stars or lightnings) Applications: controlled fusion, Plane/flame interaction...
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 dv, momentum u(t, x) := n 1 vf dv and pressure p := f (v u) 2 dv.
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)
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]
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
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
Comparison Initializations of a semi-gaussian beam in 1+1 d Solution: adaptive semi-lagrangian schemes...
Outline 1 Applications and models for charged particles The Vlasov equation Numerical methods 2 Wavelets or mesh refinement? Dynamic adaptivity The predict-and-readapt scheme
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+1t f n M n M n+1 B n (x, v) (x, v) CP, Mehrenberger, Proceedings of Cemracs 2003
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+1t f n Λ n Λ n+1 B n (x, v) (x, v) Gutnic, Haefele, Paun, Sonnendrücker, Comput. Phys. Comm. 2004
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Dynamic adaptivity See also 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) Houston, Süli, Technical report 1995, Math. Comp. 2001 Behrens, Iske, and Käser ( 1996 2006)
Prediction by recursive splitting, looking backwards M n B n Λ n B n
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 l 0 sup d γ (f ) γ l \Λ
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)
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 ): predict a first mesh M n+1 := T[B n ]M n 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 Cε-adapted to T f n Λ n is ε-adapted to f n = T[B n ]Λ n is w Cε-adapted to T f n Some papers available at http://www-irma.u-strasbg.fr/ campos