Long-time behavior of multi-step Finite Difference schemes with boundary via steepest descent and analytic combinatorics

TL;DR

This paper uses steepest descent and analytic combinatorics to analyze the long-time behavior of linear finite difference schemes, including leap-frog, near boundaries with stable or unstable conditions.

Contribution

It introduces a novel analytical approach combining steepest descent and singularity analysis to study boundary effects on finite difference schemes over long times.

Findings

Accurate description of scheme behavior near boundaries

Analysis includes stable and unstable boundary conditions

Applicable to schemes like leap-frog

Abstract

We demonstrate how steepest descent arguments and singularity analysis from analytic combinatorics allow for an accurate description of the behavior of linear numerical schemes -- including the notorious leap-frog scheme -- in presence of stable and unstable boundary conditions in the long-time limit.