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Uniformly higher order accurate schemes for dynamics of charged particles under fast oscillating magnetic fields

Megala Anandan, Benjamin Boutin, Nicolas Crouseilles

Abstract

This work deals with the numerical approximation of plasmas which are confined by the effect of a fast oscillating magnetic field (see \cite{Bostan2012}) in the Vlasov model. The presence of this magnetic field induces oscillations (in time) to the solution of the characteristic equations. Due to its multiscale character, a standard time discretization would lead to an inefficient solver. In this work, time integrators are derived and analyzed for a class of highly oscillatory differential systems. We prove the uniform accuracy property of these time integrators, meaning that the accuracy does not depend on the small parameter $\varepsilon$. Moreover, we construct an extension of the scheme which degenerates towards an energy preserving numerical scheme for the averaged model, when $\varepsilon\to 0$. Several numerical results illustrate the capabilities of the method.

Uniformly higher order accurate schemes for dynamics of charged particles under fast oscillating magnetic fields

Abstract

This work deals with the numerical approximation of plasmas which are confined by the effect of a fast oscillating magnetic field (see \cite{Bostan2012}) in the Vlasov model. The presence of this magnetic field induces oscillations (in time) to the solution of the characteristic equations. Due to its multiscale character, a standard time discretization would lead to an inefficient solver. In this work, time integrators are derived and analyzed for a class of highly oscillatory differential systems. We prove the uniform accuracy property of these time integrators, meaning that the accuracy does not depend on the small parameter . Moreover, we construct an extension of the scheme which degenerates towards an energy preserving numerical scheme for the averaged model, when . Several numerical results illustrate the capabilities of the method.

Paper Structure

This paper contains 27 sections, 13 theorems, 129 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Models
  3. Uniformly accurate numerical schemes: linear case
  4. Arbitrarily higher order accurate scheme: explicit case
  5. Uniformly accurate numerical schemes: midpoint case
  6. Midpoint schemes for the charged particles model: linear case
  7. Nonlinear system for charged particles under fast oscillating magnetic field
  8. Explicit case
  9. First order accurate scheme
  10. Second order accurate scheme
  11. Midpoint schemes for the nonlinear charged particles model
  12. SAV formulation of the averaged model
  13. Numerical scheme for the SAV formulation of the averaged model
  14. Uniformly accurate and structure preserving SAV-schemes for the charged particle model
  15. Choice 1
  16. ...and 12 more sections

Key Result

Theorem 1

Let $U(t=0) \in \mathbb{R}^d$. and $A\left(s=t/\varepsilon\right) \in {\mathcal{M}}_{d,d}(\mathbb{R})$ be a given bounded $P$-periodic function of $s$. Consider the solution $t\mapsto U(t)\in\mathbb{R}^d$ to ODEsys_E0 over a given time interval $[0,T]$. The numerical scheme Arbit order ODE_(x,q) wit for all $n=1,2,\dots,N$, with a constant $C$ independent of $\Delta t$ and $\varepsilon$.

Figures (14)

  • Figure 1: Linear case, first order explicit scheme: First order uniform accuracy with $\Delta t$.
  • Figure 2: Linear case, midpoint and UA midpoint schemes: Second order uniform accuracy with $\Delta t$.
  • Figure 3: Linear scalar case, fourth order explicit scheme: uniform accuracy with $\Delta t$.
  • Figure 4: Nonlinear case, second order explicit scheme: Second order uniform accuracy with $\Delta t$.
  • Figure 5: Nonlinear case, midpoint SAV-scheme (first choice for $b$): Second order uniform accuracy with $\Delta t$.
  • ...and 9 more figures

Theorems & Definitions (29)

  • Remark 1
  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Proposition 2
  • proof
  • Remark 2
  • ...and 19 more