Error analysis of higher order time-domain finite element methods for the one-dimensional Maxwell\u27s equations

A well known nodal discontinuous Galerkin finite element method has been extended for higher order temporal accuracy using several schemes. While common in computational fluid dynamics, less research has been conducted with these methods for computational electromagnetics. A stabilized finite element method utilizing the Streamline/Upwind Petrov-Galerkin approach is explored. This work examines several higher order temporally accurate schemes to test their viability for the Maxwell equations. Only the one-dimensional case is considered. The temporal integration methods utilized are the first two backward differentiation formula (BDF), second through fourth order modified extended backward differentiation formula (MEBDF), and second through fourth order explicit first stage singly diagonally implicit Runge- Kutta (ESDIRK) schemes. A problem using a simple Gaussian pulse to which the analytical solution is known is used to verify the desired order of accuracy. Fifth-order spatial integration using Legendre polynomials, so spatial errors will be much smaller than temporal errors