Exact stability conditions in upwinding‐scheme FDTD for the Boltzman transport equation

Abstract

SUMMARYAn in-depth stability analysis of the FDTD method under the upwinding scheme for the Boltzmann transport equation (BTE) under the relaxation time approximation is provided. Both time forward and time backward difference equations are considered. In the time forward differencing case, a sufficient stability condition is derived for the BTE with variable coefficients, and a necessary and sufficient condition is derived for the BTE with constant coefficients. In the time backward differencing case, it is shown that the differencing equations are unconditionally stable. It is shown numerically that the previously reported stability conditions in the literature are not accurate.