The Leray-Gårding method for finite difference schemes
This work provides a theoretical foundation for obtaining optimal stability estimates for finite difference approximations of hyperbolic initial boundary value problems, addressing a long-standing open problem in numerical analysis.
The authors adapt the Leray-Gårding multiplier technique to finite difference schemes for transport equations, enabling the derivation of optimal semigroup estimates for fully discrete hyperbolic initial boundary value problems, thereby solving a problem raised by Trefethen, Kreiss, and Wu.
Leray and Gårding 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 Leray and Gårding's work 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 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.