The Von Neumann Method for Stability Analysis

Various methods have been developed for the analysis of stability, nearly all of them limited to linear problems. However, even within this restriction the complete investigation of stability for initial, boundary value problems can be extremely complicated, particularly in the presence of boundary conditions and their numerical representation. The problem of stability for a linear problem with constant coefficients is now well understood when the influence of boundaries can be neglected or removed. This is the case either for an infinite domain or for periodic conditions on a finite domain. In the latter case we consider that the computational domain on the x-axis of length L is repeated periodically, and therefore all quantities, the solution, as well as the errors, can be developed in a finite Fourier series over the domain 2L. This development in the frequency domain (in space) forms the basis of the Von Neumann method for stability analysis (Sections 8.1 and 8.2). This method was developed in Los Alamos during World War II by Yon Neumann and was considered classified until its brief description in Cranck and Nic'flolson (1947) and in a publication in 1950 by Charney et at. (1950). At present this is the most widely applied technique for stability analysis, and furthermore allows an extensive investigation of the behaviour of the error as a function of the frequency content of the initial data and of the solution, as will be seen in Section 8.3. The generalization of the Yon Neumann method to multidimensional problems is presented in Section 8.4. If the problem of stability analysis can be treated generally for linear equations with constant coefficients and with periodic boundary conditions, as soon as we have to deal with nop-constant coefficients and (or) non-linear terms in the basic equations the information on stability becomes very limited. Hence we have to resort to a local stability analysis, with frozen values of the non-linear and non-constant coefficients, to make the formulation linear. In any case, linear stability is a necessary condition for non-linear problems but it is certainly not sufficient. We will touch on this difficult problem in Section 8.5. Finally, Section 8.6 presents certain general techniques in order to obtain the stability conditions from the Yon Neumann analysis.