Vol. 15, No. 1, 2020
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A third-order multirate Runge–Kutta scheme for finite volume solution of 3D time-dependent Maxwell's equations

Marina Kotovshchikova, Dmitry K. Firsov and Shiu Hong Lui
Vol. 15 (2020), No. 1, 65–87
DOI: 10.2140/camcos.2020.15.65
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.

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Keywords
multirate Runge–Kutta schemes, Maxwell's equations, three-dimensional unstructured meshes, finite volume
Mathematical Subject Classification 2010
Primary: 65L06, 65M08, 78M12
Milestones
Received: 28 October 2019
Revised: 25 February 2020
Accepted: 12 April 2020
Published: 3 June 2020
Authors
Marina Kotovshchikova
San Jose, CA United States
Dmitry K. Firsov
San Jose, CA
United States
Shiu Hong Lui
Department of Mathematics
University of Manitoba
Winnipeg, MB
Canada