Developing and analyzing fourth-order difference methods for the metamaterial Maxwell's equations

In this paper, we develop both a fourth order explicit scheme and a compact implicit scheme for solving the metamaterial Maxwell's equations. A systematic technique is introduced to prove stability and error estimate for both schemes. Numerical results supporting our analysis are presented. To our best knowledge, our convergence theory and stability results are novel, and provide the first error estimate for the fourth order finite difference methods for Maxwell's equations.