Accuracy and stability of the hyperbolic model time integration scheme revisited

TL;DR

This paper revisits the accuracy and stability of the hyperbolic model time integration scheme, clarifying its stability conditions and discussing potential issues with convergence due to stability limitations.

Contribution

It provides a detailed analysis of the stability properties of the scheme, linking operator inequalities to eigenvalue conditions, and discusses practical implications for convergence.

Findings

Stability condition matches eigenvalue magnitude requirement.

Norm of the amplification matrix can exceed one, risking convergence issues.

Practical detection and mitigation strategies are discussed.

Abstract

The hyperbolic model (HM) time integration scheme tackles parabolic problems by adding a small artificial second order time derivative term. Described by Samarskii in his 1971 book, the scheme reappeared as the generalized Du Fort-Frankel scheme in a 1976 paper by Gottlieb and Gustafsson. In this note we revisit accuracy and stability properties of the scheme. In particular, we show that the stability condition, formulated by Samarskii based on operator inequalities, coincides with the requirement that the eigenvalues of the amplification matrix (the stability function operator) are smaller than one in absolute value. However, under this condition, the norm of this matrix may exceed one and this, as recently pointed out by Corem and Ditkowski (2012), may corrupt convergence of the scheme. Hence, we also discuss whether this eventual stability lack can be detected and mitigated in practice.