Abstract. CW/ P.O. Box 94079, 1090 GB Amsterdam, The Netherlands willem@cwi.nlThis paper is concerned with the stability of numerical processes for solving initial 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). We show that the fully discretized 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 time-stepping method under consideration.A.MS subject classification: 65Ml2, 65L20, 65M70.Key words: stability of numerical processes, initial(-boundary) value problems, ordinary and partial differential equations, one-step and multistep methods, spectral collocation methods.