Summary: The Fourier method is used to analyze the dispersive, dissipative, and isotropy errors of various spatial and time discretizations applied to the Maxwell equations on multidimensional grids. Both Cartesian grids and non-Cartesian grids based on hexagons and tetradecahedra are studied and compared.
The numerical errors are quantitatively determined in terms of phase speed, wavenumber, propagation direction, gridspacings, and CFL number. The study shows that centered schemes are more efficient and accurate than upwind schemes and the non-Cartesian grids yield superior isotropy than the Cartesian ones. For the centered schemes, the staggered grids produce less errors than the unstaggered ones.
A new unstaggered algorithm which has all the best properties is introduced. Using an optimization technique to determine the nodal weights, the new algorithm provides the highest accuracy among all the schemes discussed. The study also demonstrates that a proper choice of time discretization can reduce the overall numerical errors due to the spatial discretization.