# Exponential Integrators for Stochastic Maxwell's Equations Driven by It\^o Noise

## TL;DR

This paper develops explicit exponential integrators for stochastic Maxwell's equations driven by Itô noise, achieving strong convergence orders of 1/2 and 1 depending on noise type, and preserving key physical structures in certain cases.

## Contribution

The paper introduces new exponential integrators for stochastic Maxwell's equations, with proven strong convergence orders and structure-preserving properties for additive noise cases.

## Key findings

- Strong order 1/2 convergence for general multiplicative noise
- Strong order 1 convergence for additive noise
- Exact preservation of symplectic structure and energy evolution in linear additive noise case

## Abstract

This article presents explicit exponential integrators for stochastic Maxwell's equations driven by both multiplicative and additive noises. By utilizing the regularity estimate of the mild solution, we first prove that the strong order of the numerical approximation is $\frac 12$ for general multiplicative noise. Combing a proper decomposition with the stochastic Fubini's theorem, the strong order of the proposed scheme is shown to be $1$ for additive noise. Moreover, for linear stochastic Maxwell's equation with additive noise, the proposed time integrator is shown to preserve exactly the symplectic structure, the evolution of the energy as well as the evolution of the divergence in the sense of expectation. Several numerical experiments are presented in order to verify our theoretical findings.