S-ERKN: an explicit divergence preserving method with low memory consumption for solving three-dimensional time-domain Maxwell's equations under periodic boundary conditions

Hongli Yang, Jianchen Gao and Xianyang Zeng

Journal of Electromagnetic Waves and Applications, 2026, vol. 40, issue 10, 1636-1663

Abstract: This article proposes an explicit method with low memory consumption, termed the S-ERKN (Spatial Spectral and Temporal Extended-Runge–Kutta Nyström) method, for the numerical solution of Maxwell's equations in three-dimensional time-domain electromagnetic fields with periodic boundary conditions. By discretizing spatial derivatives using the Fourier spectral method and combining them with ERKN time integrators based on operator-variation-of-constants formulas, this method achieves high-precision and high-efficiency numerical simulations of electric and magnetic fields. To address the strongly oscillatory and high-dimensional nature of Maxwell's equations with source terms, as well as the high computational cost of traditional implicit methods and the accuracy challenges of explicit methods, this paper introduces symmetric positive semi-definite operator functions $ \phi _0 $ ϕ0 and $ \phi _1 $ ϕ1, combined with the eigen-decomposition technique for a Kronecker product structure. This approach transforms high-dimensional problems into diagonal systems, significantly reducing computational complexity and memory requirements. Numerical experiments, including free-field and source-field problems, demonstrate that the method maintains high accuracy, good stability, and numerical divergence-free properties under long-term simulations and large time steps.

Date: 2026
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DOI: 10.1080/09205071.2026.2650386

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