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

zbMATH Open1481.65125arXiv2009.11657MaRDI QIDQ5018547

Publication date: 21 December 2021

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].

Full work available at URL: https://arxiv.org/abs/2009.11657

zbMATH Keywords

stability boundary conditions hyperbolic equations semigroup estimates difference approximations

Mathematics Subject Classification ID

Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) Initial value problems for first-order hyperbolic equations (35L03) Initial-boundary value problems for first-order hyperbolic equations (35L04)