Stability Estimate Based on Numerical Ranges with Application to a SpectralDiscretization of a Convection-Diffusion Equation

摘要

This paper is concerned with the stability of numerical processes for solvinginitial value problems. We present a stability result which is related to a well-known theorem by Von Neumann, but the requirements to be satisfied are less severe and easier to verify. As an illustration, we consider a simple convection-diffusion equation. For the spatial discretization we use a spectral collocation method (based on so-called Legendre-Gauss-Lobatto points) and the temporal discretization is based on a one-step method. We show that the fully discrete numerical process is stable, provided that the temporal step size is bounded by a constant depending only on the convection-diffusion equation, the number of collocation points, and the one-step method under consideration.