Fourth-Order Accurate Compact Scheme for First-Order Maxwell's Equations

We construct a compact fourth-order scheme, in space and time, for the time-dependent Maxwell's equations given as a first-order system on a staggered (Yee) grid. At each time step, we update the fields by solving positive definite second-order elliptic equations. We develop compatible boundary conditions for these elliptic equations while maintaining a compact stencil. The proposed scheme is compared computationally with a non-compact scheme and with a convolutional dispersion relation preserving (DRP) scheme.