Abstract:

In this paper, numerical methods for solving 2D nonlinear Maxwell's equations in inhomogeneous media are discussed. A multidomain Legendre-tau collocation spectral method is proposed. The proposed method is of spectral accuracy in space and second order in time. In time direction, the leap-frog Crank-Nicolson method is applied, which is a three-level scheme with the nonlinear term being treated by some collocation methods explicitly in intermediate level and the linear terms being treated by the Legendre-tau spectral method implicitly. By the implicit-explicit scheme, the numerical method is of better stability and easy implementation. We construct a reasonable weak form which can deal with the interface conditions in a way just like the natural boundary condition without any additional interface conditions. The uniform scheme without any additional interface conditions is constructed by using polynomial spaces of different degrees. For the semi-discrete and fully discrete schemes, the stability and convergence are proved, and the $L^2$-norm optimal error estimates are obtained. In numerical examples, the nonlinear terms are computed at the Chebyshev points by the fast Legendre transform to improve the efficiency of the algorithm. Numerical results show the efficiency of the proposed method for solving this nonlinear problem. Moreover, the results indicate that the spectral accuracy is achieved and not affected by the discontinuity of solutions.

Key words: 2D Maxwell's equations, Nonlinear conductivity, Inhomogeneous media, Multidomain Legendre-tau spectral method, Leap-frog Crank-Nicolson method, Optimal error estimates