A third-order multirate Runge–Kutta scheme for finite volume solution of 3D time-dependent Maxwell's equations

Abstract

A third-order multirate time-stepping based on an SSP Runge–Kutta method is applied to solve the three-dimensional Maxwell’s equations on unstructured tetrahedral meshes. This allows for an evolution of the solution on fine and coarse meshes with time steps satisfying a local stability condition to improve the computational efficiency of numerical simulations. Two multirate strategies with flexible time-step ratios are compared for accuracy and efficiency. Numerical experiments with a third-order finite volume discretization are presented to validate the theory. Our results of electromagnetic simulations demonstrate that 1D analysis is also valid for linear conservation laws in 3D. In one of the methods, significant speedup in 3D simulations is achieved without sacrificing third-order accuracy.