On Instabilities in Time Marching Methods

Abstract

We discuss in this paper the stability of the time marching (TM) method. We identify one cause of instability in the method associated with the calculation of variables involved in the convolution operation. We provide a solution to this problem, preventing the appearance of unstable poles in the Z-domain. This solution is fundamentally different from other previously presented approaches in the sense that it is not based on filtering or predictive techniques. Instead, it consists of preprocessing the known variable in the convolution equations.

An improved time marching simulation of distributed multiport networks loaded with nonlinear devices

“…The main drawback of TM is its late‐time instability, which was traditionally attributed to numerical error accumulation over time and treated with time‐averaging, use of low‐pass Finite Impulse Response (FIR) filters, or use of predictor‐corrector schemes . Nevertheless, it is shown in Becerra et al that the main cause of TM instability is the occurrence of unstable poles in deconvolution operations. These poles appear in the Z‐transform of the deconvolution kernel (usually an impulse response), and hence, the instability is solved by replacing the deconvolution kernel by its minimum phase version …”