Stability estimates based on numerical ranges with an application to a spectral method. (English) Zbl 0806.65101
Authors’ summary: This 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.
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.
Reviewer: S.Jiang (Bonn)
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 35K15 | Initial value problems for second-order parabolic equations |
Keywords:
multistep methods; stability; convection-diffusion equation; spectral collocation method; time-stepping methodReferences:
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