The Leray-Gårding method for finite difference schemes

Jean-François Coulombel

doi:10.5802/jep.25

Abstract

In [Ler53] and [ Gar56], Leray and Garding have developed a multiplier technique for deriving a priori estimates for solutions to scalar hyperbolic equations in either the whole space or the torus. In particular, the arguments in [Ler53, Gar56 ] provide with at least one local multiplier and one local energy functional that is controlled along the evolution. The existence of such a local multiplier is the starting point of the argument by Rauch in [Rau72] for the derivation of semigroup estimates for hyperbolic initial boundary value problems. In this article, we explain how this multiplier technique can be adapted to the framework of finite difference approximations of transport equations. The technique applies to numerical schemes with arbitrarily many time levels, and encompasses a somehow magical trick that has been known for a long time for the leapfrog scheme. More importantly, the existence and properties of the local multiplier enable us to derive optimal semigroup estimates for fully discrete hyperbolic initial boundary value problems, which answers a problem raised by Trefethen, Kreiss and Wu [Tre84, KW93].

Highlights

The only available general stability theory for such numerical schemes is due to Gustafsson, Kreiss and Sundström [13]
The corresponding stability estimates are restricted to zero initial data
A long standing problem in this line of research is, starting from the GKS stability estimates, which are resolvent type estimates, to incorporate nonzero initial data and to derive semigroup estimates, see, e.g., the discussion by Trefethen in [28, §4] and the conjecture by Kreiss and Wu in [17]. This problem is delicate for the following reason: the validity of the GKS stability

Summary

Introduction

Our main contribution in this article is to exhibit a suitable multiplier for the multistep recurrence relation in (1.2) With this multiplier, we can readily show that, for zero initial data, the (discrete) derivative of an energy can be controlled, as in the work by Rauch [23] on partial differential equations, by the trace estimate of (unj ) and this is where strong stability comes into play. It should be kept in mind that we assume here that each ratio ∆t/∆xj is constant, and we consider each coefficient in the operators Qσ, Bj1,σ as independent of the time and space steps This point of view has some technical advantages since we may for instance view the Qσ’s as bounded operators with norms that are independent of the time and space steps, and all estimates in Theorem 1 are independent (1)It would even give the claim of Theorem 1 for nonzero initial data provided that the nonglancing condition of [4] is satisfied, but we do not wish to make such an assumption here. We postpone the extension of this work to parabolic or dispersive equations to a future work

Examples

The energy-dissipation balance for finite difference schemes

Semigroup estimates for fully discrete initial boundary value problems

Conclusion and perspectives