Stability of finite difference schemes for hyperbolic initial boundary value problems: numerical boundary layers
This work provides a theoretical foundation for analyzing stability of finite difference schemes for hyperbolic problems, extending semigroup estimates to a broader class of numerical methods.
The paper develops a unified theory for boundary layer expansions in discretized transport equations with Dirichlet boundary conditions, enabling discrete semigroup estimates for numerical schemes with arbitrarily many time levels, which were previously limited to two time levels.
In this article, we give a unified theory for constructing boundary layer expansions for dis-cretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels, while semigroup estimates were restricted, up to now, to numerical schemes with two time levels only.