Investigation on polynomial integrators for time-domain electromagnetics using a high-order discontinuous Galerkin method

In this work, we investigate the application of polynomial integrators in a high-order discontinuous Galerkin method for solving the time-domain Maxwell equations. After the spatial discretization, we obtain a time-continuous system of ordinary differential equations of the form, ∂tY(t)=HY(t)∂tY(t)=HY(t), where Y(t)Y(t) is the electromagnetic field, HH is a matrix containing the spatial derivatives, and t   is the time variable. The formal solution is written as the exponential evolution operator, exp(tH)exp(tH), acting on a vector representing the initial condition of the electromagnetic field. The polynomial integrators are based on the approximation of exp(tH)exp(tH) by an expansion of the form ∑m=0∞gm(t)Pm(H), where gm(t)gm(t) is a function of time and Pm(H)Pm(H) is a polynomial of order m   satisfying a short recursion. We introduce a general family of expansions of exp(tH)exp(tH) based on Faber polynomials. This family of expansions is suitable for non-Hermitian matrices, and consequently the proposed integrators can handle absorbing media and conductive materials. We discuss the efficient implementation of this technique, and based on some test problems, we compare the virtues and shortcomings of the algorithm. We also demonstrate how this scheme provides an efficient alternative to standard explicit integrators.