Stability Analysis of Combined Compact Difference Method

We investigate the stability of a numerical method using the Combined Compact Difference (sp-CCD) method with spectral-like resolution for a 1-D advection diffusion problem. The sp-CCD method was proposed by Nihei and Ishii and has high-accuracy and high-resolution. The finite difference representation of the equation consists of the sp-CCD method for the spatial derivatives and the Runge-Kutta (RK) method for the time marching. In order to investigate the influence of the boundary conditions we use the matrix method in the stability analysis. It is shown that the numerical method with some parameters is stable in the problem with periodic boundaries, but it is unstable in the problem with nonperiodic boundaries. We also propose a new stable boundary scheme for the problem with nonperiodic boundary.