Convergence Analysis of Fourth-Order Runge–Kutta FDTD Method for Maxwell’s Equations

Abstract In this paper, we present a rigorous stability analysis and error estimate for an explicit fourth-order Runge–Kutta (RK) time-integration scheme coupled with finite-difference time-domain (FDTD) spatial discretization, applied to the time-dependent Maxwell’s equations in free space. While numerous RK-FDTD methods have been successfully developed and applied for electromagnetic simulations, a rigorous mathematical analysis specifically for this class of schemes has remained open in the literature. To the best of our knowledge, this work constitutes the first theoretical effort to bridge this gap by establishing a comprehensive stability analysis and error estimate for the fourth-order RK-FDTD method. Our theoretical results are supported by some numerical experiments conducted for the three-dimensional Maxwell’s equations.