Solving Time Domain Helmholtz Wave Equation with Mod-fdm

In this work, we present a marching-on in degree finite difference method (MOD-FDM) to solve the time domain Helmholtz wave equation. This formulation includes electric and magnetic current densities that are expressed in terms of the incident field for scattering problems for an open region to implement a plane wave excitation. The unknown time domain functional variations for the electric field are approximated by an orthogonal basis function set that is derived using the Laguerre polynomials. These temporal basis functions are also used to expand current densities. With the representation of the derivatives of the time domain variable in an analytic form, all the time derivatives of the field and current density can be handled analytically. By applying a temporal testing procedure, we get a matrix equation that is solved in a marching-on in degree technique as the degree of the temporal basis functions is increased. Numerical results computed using the proposed formulation are presented and compared with the solutions of the conventional time domain finite difference method (TD-FDM) and analytic solutions.