New Taylor-expansion method for solving a general class of wave equations

Abstract

We derive a general method for solving second-order wave equations on the basis of a fourth-order Taylor expansion, with respect to time, of the field and its time derivative. The method requires explicit evaluation of space derivatives up to fourth order of the field and its time derivative, and the time step is subject to a stability criterion, similar to the one routinely applied in conjunction with the time-domain–finite-difference method. The Taylor-series method is generalizable to vector wave equations, to wave equations involving dissipation, and to the Helmholtz wave equation. The method is equally applicable to situations involving large or small variations in the refractive index. Sample numerical solutions in one and two space dimensions are described.