Kernel Analysis of the Discretized Finite Difference and Finite Element Shallow-Water Models

Abstract

A constructive linear algebra approach is developed to characterize the kernels of the discretized shallow-water equations. Three kernel relations are identified as necessary conditions for the discretized system to share the same stationary properties as the continuous system. This matrix kernel scheme is computed using MATLAB and applied to investigate the presence, number, and structure of spurious modes arising in typical finite difference and finite element schemes. The kernel concept is then used to characterize the smallest representable vortices for several representative discrete finite difference and finite element schemes. Both uniform and unstructured mesh situations are considered and compared. Numerical experiments are consistent with the analytic results.