Instability in Leapfrog and Forward–Backward Schemes

Abstract

This paper shows that in the linearized shallow-water equations, the numerical schemes can become weakly unstable for the 2?x wave in the C grid when the Courant number is 1 in the forward?backward scheme and 0.5 in the leapfrog scheme because of the repeated eigenvalues in the matrices. The instability can be amplified and spread to other waves and smaller Courant number if the diffusion term is included. However, Shuman smoothing can control the instability.