Stability analysis of the numerical Method of characteristics applied to a class of energy-preserving hyperbolic systems. Part II: Nonreflecting boundary conditions

• Published in: Journal of computational and applied mathematics 2019-08, Vol.356, p.267-292
• Main Authors: Lakoba, T.I., Deng, Z.
• Format: Article
• Language: English
• Subjects: Block-Toeplitz matrices; Coupled-wave equations; Method of characteristics; Non-periodic boundary conditions; Numerical instability
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2019.01.042

We show that imposition of non-periodic, in place of periodic, boundary conditions (BC) can alter stability of modes in the Method of characteristics (MoC) employing certain ordinary-differential equation (ODE) numerical solvers. Thus, using non-periodic BC may render some of the MoC schemes stable for most practical computations, even though they are unstable for periodic BC. This fact contradicts a statement, found in some textbooks and known as part of the Babenko–Gelfand criterion, that an instability detected by the von Neumann analysis for a given numerical scheme implies an instability of that scheme with arbitrary (i.e., non-periodic) BC. We explain the mechanism behind this contradiction, which lies in a certain property of the scheme’s eigenmodes that is assumed by the Babenko–Gelfand criterion but does not hold for eigenmodes of some of the MoC-based schemes. We also show that, and explain why, for the MoC employing some other ODE solvers, stability of the modes may indeed not be improved by non-periodic BC, as the Babenko–Gelfand criterion implies.