Stable FDTD for the Time-Domain Wave Equation Resulted from Spherical Harmonic Expansions

In this paper, the solution of the time-domain wave equation involved in spherical harmonic expansion (SHE) is revisited. The finite difference time domain (FDTD) method is a preferred choice for the second-order time-domain wave equation, while the traditional Courant–Friedrich–Levy (CFL) condition cannot be applied directly. It is found that the stability condition of the wave equation is associated with the degree
$n$
. As the degree
$n$
increases, the upper limit of the time step declines. However, due to the strictness of the derived stability condition, we opted to verify it using non-uniform spatial sampling. Numerical results obtained from this approach provide justification for our analysis.