On the edge of stability analysis

The application of high order methods to solve problems with physical boundary conditions in many cases requires a careful treatment near the boundary, where additional numerical boundary schemes have to be introduced. The choice of boundary schemes influences most of the times the stability of the numerical method. The von Neumann analysis does not allow us to define accurately the influence of boundary conditions on the stability of the scheme. The spectral analysis, often called the matrix method, considers the eigenvalues of the matrix iteration of the scheme and although they reflect some of the influence of boundary conditions on the stability, many times eigenvalues fail to capture the transient effects in time-dependent partial differential equations. The Lax stability 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.