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Splitting integrators for linear Vlasov equations with stochastic perturbations

Charles-Edouard Bréhier, David Cohen

Abstract

We consider a class of linear Vlasov partial differential equations driven by Wiener noise. Different types of stochastic perturbations are treated: additive noise, multiplicative Itô and Stratonovich noise, and transport noise. We propose to employ splitting integrators for the temporal discretization of these stochastic partial differential equations. These integrators are designed in order to preserve qualitative properties of the exact solutions depending on the stochastic perturbation, such as preservation of norms or positivity of the solutions. We provide numerical experiments in order to illustrate the properties of the proposed integrators and investigate mean-square rates of convergence.

Splitting integrators for linear Vlasov equations with stochastic perturbations

Abstract

We consider a class of linear Vlasov partial differential equations driven by Wiener noise. Different types of stochastic perturbations are treated: additive noise, multiplicative Itô and Stratonovich noise, and transport noise. We propose to employ splitting integrators for the temporal discretization of these stochastic partial differential equations. These integrators are designed in order to preserve qualitative properties of the exact solutions depending on the stochastic perturbation, such as preservation of norms or positivity of the solutions. We provide numerical experiments in order to illustrate the properties of the proposed integrators and investigate mean-square rates of convergence.
Paper Structure (24 sections, 8 theorems, 90 equations, 16 figures)

This paper contains 24 sections, 8 theorems, 90 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries on the deterministic linear Vlasov equation
  4. Analysis and properties of the problem
  5. Numerical approximation
  6. The stochastic linear Vlasov equation perturbed by additive noise
  7. Analysis and properties of the problem
  8. Splitting scheme
  9. Numerical experiments
  10. The stochastic linear Vlasov equation perturbed by multiplicative noise
  11. Itô interpretation
  12. Analysis and properties of the problem
  13. Splitting scheme
  14. Numerical experiments
  15. Stratonovich interpretation
  16. ...and 9 more sections

Key Result

Proposition 1

Assume that $\sigma_k\in L_{x,v}^2$ for all $1\le k\le K$, and that $f_0\in L_{x,v}^2$. Let $\bigl(f^{\rm add}(t)\bigr)_{t\ge 0}$ be the solution of the SPDE eq:SPDE-add driven by the additive noise eq:Wtxv. Then for all $t\ge 0$ one has $f^{\rm add}(t)\in L^2(\Omega, L_{x,v}^2)$ and

Figures (16)

  • Figure 1: Snapshots: approximation of the solution of the deterministic PDE \ref{['eq:Vlasov']} with initial value $f_0$ given by \ref{['eq:f0']}, at times $\{0,0.5,1,1.5,2,2.5\}$, using the Lie--Trotter splitting scheme \ref{['eq:LTdeter']} with time-step size $\tau=0.1$.
  • Figure 2: Snapshots: approximation of the solution of the stochastic PDE with additive noise \ref{['eq:SPDE-add']} with initial value $f_0$ given by \ref{['eq:f0']}, with $\sigma_1$ given by \ref{['eq:sigma1cos']} at times $\{0,0.5,1,1.5,2,2.5\}$, using the Lie--Trotter splitting scheme \ref{['eq:LT-add']} with time-step size $\tau=0.1$.
  • Figure 3: Snapshots: approximation of the solution of the stochastic PDE with additive noise \ref{['eq:SPDE-add']} with initial value $f_0$ given by \ref{['eq:f0']}, with $\sigma_1$ given by \ref{['eq:sigma1sin']} at times $\{0,0.5,1,1.5,2,2.5\}$, using the Lie--Trotter splitting scheme \ref{['eq:LT-add']} with time-step size $\tau=0.1$.
  • Figure 4: Trace formula: illustration of Proposition \ref{['propo:tracenum']} when applying the Lie--Trotter splitting scheme \ref{['eq:LT-add']} to the SPDE with additive noise \ref{['eq:SPDE-add']} with time-step size $\tau=0.1$.
  • Figure 5: Mean-square errors: Lie--Trotter scheme \ref{['eq:LT-add']} applied to the SPDE with additive noise \ref{['eq:SPDE-add']} driven by one-dimensional noise ($K=1$, left) and two-dimensional noises ($K=2$, right).
  • ...and 11 more figures

Theorems & Definitions (22)

  • Proposition 1
  • proof : First proof of Proposition \ref{['propo:trace']}
  • proof : Second proof of Proposition \ref{['propo:trace']}
  • Proposition 2
  • proof : First proof of Proposition \ref{['propo:tracenum']}
  • proof : Second proof of Proposition \ref{['propo:tracenum']}
  • Proposition 3
  • proof : Proof of Proposition \ref{['propo:multIto']}
  • Remark 4
  • Proposition 5
  • ...and 12 more