Chebyshev Spectral Methods for Limited-Area Models. Part II: Shallow Water Model

Abstract

Chebyshev spectral methods were studied in Part I for the linear advection equation in one dimension. Here we extend these methods to the nonlinear shallow water equations in two dimensions. Numerical models are constructed for a limited domain on a β-plane, using open (characteristic) boundary conditions based on Rieman invariants to simulate an unbounded domain. Reflecting boundary conditions (wall and balance) are also considered for comparison. We discuss the formulation of the Chebyshev–tau and Chebyshev–collocation discretizations for this problem. The tau discretization avoids aliasing error in evaluating quadratic nonlinear terms, while the collocation method is simpler to program. Numerical results from a linearized one-dimensional test problem demonstrate that with the characteristic boundary conditions the stability properties for various explicit time differencing schemes an essentially the same as obtained in Part I for the linear advection equation. These open boundary conditions also give much more accurate results than the reflecting boundary conditions. In two dimensions, numerical results from the nonlinear models indicate that the Chebyabev–tau discretization should be based on the rotational form of the equations for efficiency, while the Chebyshev–collocation discretization should be based on the advective form for accuracy. Little difference is seen between the tau and collocation solutions for the test cases considered, other than efficiency: with explicit time differencing, the collocation model requires an order of magnitude less computer time.