Parareal Algorithms for Stochastic Maxwell Equations Driven by Multiplicative Noise

TL;DR

This paper develops and analyzes parareal algorithms for stochastic Maxwell equations with multiplicative noise, demonstrating their convergence, efficiency, and improved performance over traditional methods through theoretical and numerical validation.

Contribution

It introduces parareal algorithms using stochastic exponential integrators for stochastic Maxwell equations, with proven convergence rates and enhanced computational efficiency.

Findings

Convergence rates are proportional to iteration number k/2.

Larger damping coefficients σ accelerate convergence.

Algorithms outperform traditional exponential methods in long-term simulations.

Abstract

This paper investigates the parareal algorithms for solving the stochastic Maxwell equations driven by multiplicative noise, focusing on their convergence, computational efficiency and numerical performance. The algorithms use the stochastic exponential integrator as the coarse propagator, while both the exact integrator and the stochastic exponential integrator are used as fine propagators. Theoretical analysis shows that the mean square convergence rates of the two algorithms selected above are proportional to $k/2$, depending on the iteration number of the algorithms. Numerical experiments validate these theoretical findings, demonstrating that larger iteration numbers $k$ improve convergence rates, while larger damping coefficients $\sigma$ accelerate the convergence of the algorithms. Furthermore, the algorithms maintain high accuracy and computational efficiency, highlighting their significant advantages over traditional exponential methods in long-term simulations.