Long-time behavior of multi-step Finite Difference schemes with boundary via steepest descent and analytic combinatorics
This work addresses numerical stability issues in computational methods for researchers in applied mathematics and scientific computing, though it appears incremental as it applies known techniques to a specific problem.
The paper tackled the long-time behavior of multi-step finite difference schemes with boundary conditions, using steepest descent and analytic combinatorics to accurately describe the behavior of linear numerical schemes like the leap-frog scheme under stable and unstable boundary conditions.
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.