Tamed stability of finite difference schemes for the transport equation on the half-line

Abstract

In this paper, we prove that, under precise spectral assumptions, some finite difference approximations of scalar leftgoing transport equations on the positive half-line with numerical boundary conditions areℓ1\ell ^1-stable butℓq\ell ^q-unstable for anyq>1q>1. The proof relies on the accurate description of the Green’s function for a particular family of finite rank perturbations of Toeplitz operators whose essential spectrum belongs to the closed unit disk and with a simple eigenvalue of modulus11embedded into the essential spectrum.