A discontinuous Galerkin method for solving transient Maxwell equations with nonlinear material properties

Discontinuous Galerkin time-domain method (DGTD) has been used extensively in computational electromagnetics for analyzing transient electromagnetic wave interactions on structures described with linear constitutive relations. DGTD expands unknown fields independently on disconnected mesh elements and uses numerical flux to realize information exchange between fields on different elements (J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Method, 2008). The numerical flux of choice for `linear´ Maxwell equations is the upwind flux, which mimics accurately the physical behavior of electromagnetic waves on discontinuous boundaries. It is obtained from the analytical solution of the Riemann problem defined on the boundary of two neighboring mesh elements.