An operator splitting Legendre-tau spectral method for Maxwell’s equations with nonlinear conductivity in two dimensions

Abstract

In the paper, an operator splitting Legendre-tau spectral method is proposed to solve Maxwell’s equations with nonlinear conductivity in two dimensions. By the implicit–explicit numerical scheme, the linear and nonlinear parts are treated in different ways. The linear terms are treated by the Legendre-tau spectral method implicitly, and the nonlinear terms are treated by some collocation methods explicitly. The scheme uses polynomial spaces of different degrees to approximate the electric and magnetic fields, respectively, so that they can be decoupled in computation. The splitting leap frog Crank–Nicolson method is applied in time discretization, which is a three-level scheme with the nonlinear term being treated explicitly in intermediate level. With the application of inverse inequality, for the Legendre interpolation operator, optimal L2-error estimates are achieved for the proposed scheme. In numerical examples, the nonlinear terms are computed at Chebyshev–Gauss–Lobatto points by the fast Legendre transform. Numerical results validate the efficiency and spectral accuracy of the scheme.