The Leray-G{\aa}rding method for finite difference schemes. II. Smooth crossing modes

TL;DR

This paper extends the Leray-G{ {a}}rding method to finite difference schemes for hyperbolic PDEs, allowing for crossing modes and providing energy estimates crucial for stability analysis.

Contribution

It generalizes previous energy-dissipation constructions to cases with crossing modes, enhancing the analysis of finite difference schemes for hyperbolic problems.

Findings

Constructed energy and dissipation functionals with crossing modes

Derived semigroup estimates for fully discrete hyperbolic problems

Extended stability analysis beyond separated modes

Abstract

In [Cou15] a multiplier technique, going back to Leray and G{\aa}rding for scalar hyperbolic partial differential equations, has been extended to the context of finite difference schemes for evolutionary problems. The key point of the analysis in [Cou15] was to obtain a discrete energy-dissipation balance law when the initial difference operator is multiplied by a suitable quantity. The construction of the energy and dissipation functionals was achieved in [Cou15] under the assumption that all modes were separated. We relax this assumption here and construct, for the same multiplier as in [Cou15], the energy and dissipation functionals when some modes cross. Semigroup estimates for fully discrete hy-perbolic initial boundary value problems are deduced in this broader context by following the arguments of [Cou15].