Faber Polynomial Based Approximations of Nonlinear Integrators for Electrodynamics

Faber polynomial based approximations of time dependent exponential integrators for the numerical solution of Maxwell's equations based on explicit algorithms are extended for the inclusion of nonlinear media so that multiphysical problems can be considered. For this purpose, appropriate exponential integrators, such as Lawson type and Rosenbrock type integrators are reviewed and investigated. The formulation with regard to Rosenbrock type integrators is extended to develop exponential integrators allowing the straightforward implementation of the resulting explicit algorithm and the application of large time step sizes opening up the potential for a parallel implementation. The Faber polynomial based approach for the approximation of the exponential integrator is utilized and as in the case of linear material models has an order of magnitude higher efficiency than comparable conventional methods, especially, when a high accuracy is required.