High Order Algorithms for Maxwell Equations in Complex Geometries


A fourth order embedded boundary method for solving the Maxwell equations in geometrically complex domain has been developed. The numerical scheme is fourth order in both space and time. The spatial discretization method utilizes the main idea of the well knows second order embedded boundary algorithm, namely the assignment of some variables to cell centers of both regular grid cells and cells cut by the interface, and the finite volume integration of the other variables near the interface using the interface constraints. The forth order accuracy is achieved by building of high order mappings between cell averages and point values for the magnetic field in cut cells, and high order polynomial representations of the electric field along the grid lines. The time integration is based on Yoshida's method of building symplectic higher order schemes from lower order ones, which in our case are the second order symplectic leapfrog integrators. The algorithm, its numerical implementation, and numerical tests demonstrating the convergence and computational cost will be presented and compared with other existing schemes. The conservative (symplectic) property of Yoshida scheme and other temporally high order schemes will also be compared. The main motivation for the algorithm development is to improve solutions of Maxwell's equations in geometrically complex domains and to avoid numerical difficulties typical for unstructured finite element schemes for the simulations of electromagnetic waves coupled with charges particles.