The Positive-definite Stability Analysis for Marching-on-in-time Schemes

Abstract

The positive-definite stability analysis (PDSA) is presented as a technique complementary to the companion-matrix stability analysis (CMSA). The PDSA is used to analyze the stability of marching-on-in-time (MOT) schemes. The heart of the PDSA is formed by the analysis on particular linear combinations of interaction matrices from an MOT scheme, which are assumed to be real-valued. If these are all positive definite, then the PDSA guarantees the stability of the scheme. The PDSA can be of a lower complexity than the full CMSA. The construction of the PDSA is shown and applied to two numerical examples.