Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Long-time weak convergence analysis of a semi-discrete scheme for stochastic Maxwell equations

Chuchu Chen, Jialin Hong, Ge Liang

TL;DR

This work establishes the first long-time weak convergence result for a numerical scheme solving stochastic Maxwell equations with multiplicative noise. By transforming the Kolmogorov equation and constructing an adapted auxiliary process, the semi-implicit Euler scheme is shown to attain a weak order of $1$ (twice its strong order $1/2$). The authors derive convergence rates for numerical invariant measures, prove time-average limit theorems (SLLN and CLT) for the time-averaged numerical solution, and provide a convergence rate for the multi-level Monte Carlo estimator. These results substantiate the long-time statistical accuracy of the scheme and offer practical guidance for ergodic simulations of stochastic Maxwell dynamics.

Abstract

It is known from the monograph [1, Chapter 5] that the weak convergence analysis of numerical schemes for stochastic Maxwell equations is an unsolved problem. This paper aims to fill the gap by establishing the long-time weak convergence analysis of the semi-implicit Euler scheme for stochastic Maxwell equations. Based on analyzing the regularity of transformed Kolmogorov equation associated to stochastic Maxwell equations and constructing a proper continuous adapted auxiliary process for the semi-implicit scheme, we present the long-time weak convergence analysis for this scheme and prove that the weak convergence order is one, which is twice the strong convergence order. As applications of this result, we obtain the convergence order of the numerical invariant measure, the strong law of large numbers and central limit theorem related to the numerical solution, and the error estimate of the multi-level Monte Carlo estimator. As far as we know, this is the first result on the weak convergence order for stochastic Maxwell equations.

Long-time weak convergence analysis of a semi-discrete scheme for stochastic Maxwell equations

TL;DR

This work establishes the first long-time weak convergence result for a numerical scheme solving stochastic Maxwell equations with multiplicative noise. By transforming the Kolmogorov equation and constructing an adapted auxiliary process, the semi-implicit Euler scheme is shown to attain a weak order of (twice its strong order ). The authors derive convergence rates for numerical invariant measures, prove time-average limit theorems (SLLN and CLT) for the time-averaged numerical solution, and provide a convergence rate for the multi-level Monte Carlo estimator. These results substantiate the long-time statistical accuracy of the scheme and offer practical guidance for ergodic simulations of stochastic Maxwell dynamics.

Abstract

It is known from the monograph [1, Chapter 5] that the weak convergence analysis of numerical schemes for stochastic Maxwell equations is an unsolved problem. This paper aims to fill the gap by establishing the long-time weak convergence analysis of the semi-implicit Euler scheme for stochastic Maxwell equations. Based on analyzing the regularity of transformed Kolmogorov equation associated to stochastic Maxwell equations and constructing a proper continuous adapted auxiliary process for the semi-implicit scheme, we present the long-time weak convergence analysis for this scheme and prove that the weak convergence order is one, which is twice the strong convergence order. As applications of this result, we obtain the convergence order of the numerical invariant measure, the strong law of large numbers and central limit theorem related to the numerical solution, and the error estimate of the multi-level Monte Carlo estimator. As far as we know, this is the first result on the weak convergence order for stochastic Maxwell equations.
Paper Structure (12 sections, 17 theorems, 144 equations)

This paper contains 12 sections, 17 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Properties of the solution of stochastic Maxwell equations
  3. Temporal semi-discretization
  4. Properties of the numerical solution
  5. Long-time weak convergence analysis
  6. Kolmogorov equation
  7. Weak convergence order
  8. Applications
  9. Convergence order between invariant measures $\pi$ and $\pi^{\Delta t}$
  10. Limit theorems related to the time average of the numerical solution
  11. Convergence order of multi-level Monte Carlo estimator
  12. The Proof of Lemma \ref{['sfu4']}

Key Result

Proposition 2.3

Let $r\geq 2$, $u_0\in L^r(\Omega;\mathbb{H})$, and Assumption ass with $\rho=0$ hold. If $\sigma_0>2\widetilde{\alpha}_{F,0}+2(r-1)\widetilde{\alpha}_{B,0}^2$, then where the positive constant $C$ depends on $\sigma_0,r,\widetilde{\alpha}_{F,0}$ and $\widetilde{\alpha}_{B,0}$.

Theorems & Definitions (28)

  • Remark 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • Proposition 2.7
  • Proposition 3.1
  • proof
  • ...and 18 more