On the edge of stability analysis

Abstract

The application of high order methods to solve problems with physical boundary conditions in many cases implies to consider a different numerical approximation on the discrete points near the boundary. The choice of these approximations, called the numerical boundary conditions, influence most of the times the stability of the numerical method. Some theoretical analysis for stability, such as the von Neumann analysis, do not take into account the influence of the numerical representation of the boundary conditions on the overall stability of the scheme. The spectral analysis considers the eigenvalues of the matrix iteration of the scheme and although they reflect some of the influence of numerical boundary conditions on the stability, many times eigenvalues fail to capture the transient effects in time-dependent partial differential equations. The Lax analysis does provide information on the influence of numerical boundary conditions although in practical situations it is generally not easy to derive the corresponding stability conditions. In this paper we present properties that relates the von Neumann analysis, the spectral analysis and the Lax analysis and show under which circumstances the von Neumann analysis together with the spectral analysis provides sufficient conditions to achieve Lax stability.