Fully discrete hyperbolic initial boundary value problems with nonzero initial data
For researchers in numerical analysis of hyperbolic PDEs, this provides a more general stability theory, though it is an incremental extension of known results.
The paper extends maximal stability estimates for non-glancing finite difference approximations of hyperbolic initial boundary value problems from zero to nonzero initial data in ℓ², covering schemes with arbitrary time levels.
The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the case of zero initial data. In this article, we consider the class of non-glancing finite difference approximations to the hyperbolic operator. We show that the maximal stability estimates that are known for zero initial data and nonzero boundary source term extend to the case of nonzero initial data in â 2 . The main novelty of our approach is to cover finite difference schemes with an arbitrary number of time levels. As an easy corollary of our main trace estimate, we recover former stability results in the semigroup sense by Kreiss [Kre68] and Osher [Osh69b].