Alternating Direction Implicit Methods for FDTD Using the Dey-Mittra Embedded Boundary Method

Abstract

The alternating direction implicit (ADI) method is an attractive option to use in avoiding the Courant- Friedrichs-Lewy (CFL) condition that limits the size of the time step required by explicit finite-difference time-domain (FDTD) methods for stability. Implicit methods like Crank-Nicholson offer the same advantages as ADI methods but they do not rely on simple, one-dimensional, tridiagonal system solves for which there are well-known fast solution methods. To date, the ADI method applied to the FDTD method for curved domains has been used within the context of subgridding (i.e., local refinement) or for stairstepped boundaries that are only first-order accurate. A popular second- order accurate approach to representing smooth domains with the FDTD method is the Dey-Mittra embedded boundary method. However, to be useful in a realistic setting, the cells with only a small fraction of their volume inside the domain need to be discarded from simulations for stability considerations or else the time step size will be prohibitively small. Using the ADI method instead of the explicit method implies that time step can be chosen to depend on accuracy and no cells need discarding. We show in this paper the ability to maintain stability beyond the CFL limit for the Dey-Mittra method without discarding any cells. We also consider convergence of the ADI method as compared to the standard explicit method that is limited by the CFL condition.