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 of 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 of 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.