An ADI-Yee's scheme for Maxwell's equations with discontinuous coefficients

Highlights: • A new method for solving Maxwell equations with discontinuities and source terms. • The new method preserves Yee's method and is second order accurate in time and space. • Non-trivial stability and convergence analysis via eigenvalues estimates. • Non-trivial examples in 1D, 2D, and scattering simulations are presented. An alternating directional implicit (ADI)-Yee's scheme is developed for Maxwell's equations with discontinuous material coefficients along one or several interfaces. In order to use Yee's scheme with the presence of discontinuities, some intermediate quantities along the interface are introduced. The intermediate quantities are from the solutions and their derivatives on the interface and should satisfy some interface conditions. In discretization, those quantities are actually determined implicitly. For a fixed interface and a fixed time step size, the linear system of equations for the intermediate quantities can be pre-determined, so is the PLU or SVD decomposition of the coefficient matrix of the linear system. The ADI-Yee's scheme maintains the structure (the finite difference scheme with modified right-hand sides) as well as the accuracy and stability of Yee's scheme even with the presence of discontinuities. Theoretical analysis and numerical examples are also provided.