2025/04/01 by Romain Bonnet-Eymard, Bonnet-Eymard, Romain, Jean-François Coulombel +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2504.00667
openalex publication_date 2025/04/01 · openalex created_date 2025/10/09 · openalex updated_date 2026/07/28
The purpose of this note is to investigate the coupling of Dirichlet and Neumann numerical boundary conditions for the transport equation set on an interval. When one starts with a stable finite difference scheme on the lattice ℤ and each numerical boundary condition is taken separately with the Neumann extrapolation condition at the outflow boundary, the corresponding numerical semigroup on a half-line is known to be bounded. It is also known that the coupling of such numerical boundary conditions on a compact interval yields a stable approximation, even though large time exponentially growing modes may occur. We review the different stability estimates associated with these numerical boundary conditions and give explicit examples of such exponential growth phenomena for finite difference schemes with ''small'' stencils. This provides numerical evidence for the optimality of some stability estimates on the interval.