Adaptive semi-Lagrangian schemes for transport (how to predict - - PowerPoint PPT Presentation

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

1

Motivation Applications and models for charged particles The Vlasov equation Numerical methods

2

Adaptive semi-Lagrangian schemes Wavelets or mesh refinement ? Dynamic adaptivity The predict-and-readapt scheme

Motivation Adaptive semi-Lagrangian schemes

Outline

1

Motivation Applications and models for charged particles The Vlasov equation Numerical methods

2

Adaptive semi-Lagrangian schemes 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 {xi(t), vi(t)}i≤N by a distribution density f (t, x, v)

binary collisions Bolztmann equation mean-field approximation Vlasov equation ∂tf (t, x, v) + v ∂xf (t, x, v) + F(t, x, v) ∂vf (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.

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

{(xi(t), vi(t)) : i ≤ N} {fi(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

{(xi(t), vi(t)) : i ≤ N}

Principle: approach the density distribution f by transporting sampled "macro-particles"

initialization: deterministic approximation of f0 macro-particles {xi(0), vi(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

{fi(t) : i ≤ N}

Principle: use a transport-interpolation scheme

initialization: projection of f0 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

1

Motivation Applications and models for charged particles The Vlasov equation Numerical methods

2

Adaptive semi-Lagrangian schemes Wavelets or mesh refinement ? Dynamic adaptivity The predict-and-readapt scheme

Motivation Adaptive semi-Lagrangian schemes

Adaptive semi-Lagrangian scheme (1): adaptive meshes

Knowing fn ≈ f (tn := n∆t ), approach the backward flow B(tn) : (x, v) → (X, V )(tn; tn+1, x, v) by a diffeomorphism Bn = B[fn] transport the numerical solution with T : fn → fn ◦ Bn then interpolate on the new mesh Mn+1: fn+1 := PMn+1T fn

M n M n+1 (x, v) Bn(x, v)

CP, Mehrenberger, Proceedings of Cemracs 2003

Motivation Adaptive semi-Lagrangian schemes

Adaptive semi-Lagrangian scheme (2): wavelets

Knowing fn ≈ f (tn := n∆t ), approach the backward flow B(tn) : (x, v) → (X, V )(tn; tn+1, x, v) by a diffeomorphism Bn = B[fn] transport the numerical solution with T : fn → fn ◦ Bn then interpolate on the new grid Λn+1: fn+1 := PΛn+1T fn

Λn Λn+1 Bn(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 Mn, resp. Λn, well-adapted to fn, predict Mn+1, resp. Λn+1, well-adapted to T fn 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

Λn M n Bn Bn

Motivation Adaptive semi-Lagrangian schemes

Interpolation error estimates

hierarchical conforming spaces build on quad meshes and the corresponding P1 interpolation PM satisfies (I − PM)f L∞ sup

α∈M

|f |W 2,1(α) the wavelet interpolation PΛ : f →

γ∈Λ dγ(f )ϕγ satisfies

(I − PΛ)f L∞

• ℓ≥0

sup

γ∈∇ℓ\Λ

|dγ(f )|

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 (Mn, fn): ⋄ predict a first mesh ˜ Mn+1 := T[Bn]Mn ⋄ perform semi-Lagrangian scheme ˜ fn+1 := P ˜

Mn+1T fn

⋄ then re-adapt the mesh Mn+1 := Aε(˜ fn+1) ⋄ and interpolate again fn+1 := PMn+1˜ fn+1 Theorem (CP, Mehrenberger) Low-cost prediction algorithms satisfy: Mnis ε-adapted to fn = ⇒ T[Bn]Mnis w Cε-adapted to T fn Λn is ε-adapted to fn = ⇒ T[Bn]Λnis w Cε-adapted to T fn Some papers available at http://www-irma.u-strasbg.fr/ campos