High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations

In this paper, we motivate the use of high-order integration methods for finite element solutions of the time dependent Maxwell equations.. In particular, we present a symplectic algorithm for the integration of the coupled first-order Maxwell equations for computing the time dependent electric and magnetic fields. Symplectic methods have the benefit a conserving total electromagnetic field energy and are, therefore, preferred over dissipative methods (such as traditional Runge-Kutta) in applications that require high-accuracy and energy conservation over long periods of time integration. We show that in the context or symplectic methods, several popular schemes can be elegantly cast in a single algorithm. We conclude with some numerical examples which demonstrate the superior performance of high-order time integration methods.