The Leray-Gårding method for finite difference schemes. II.

Smooth crossing modes

Abstract

In Coulombel (2015) a multiplier technique, going back to Leray and Gård- ing 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 Coulombel (2015) was to obtain a discrete energy-dissipation balance law when the initial difference operator is multiplied by a suitable quan- tity (the so-called multiplier). The construction of the energy and dissipation functionals was achieved in Coulombel (2015) under the assumption that all modes were separated. We relax this assumption here and construct, for the same multiplier as in Coulombel (2015), the energy and dissipation functionals when some modes cross. Semigroup estimates for fully discrete hyperbolic initial boundary value problems are deduced in this broader context.