Numerical solution of the shallow water equations with a fractional step method

A fractional step technique for the numerical solution of the shallow water equations is applied to study the evolution of the potential vorticity field. The height and velocity field of the shallow water equations are discretized on a fixed Eulerian grid and time-stepped with a fractional step method recently reported in M. Shoucri, Comput. Phys. Comm. 164 (2004) 396; M. Shoucri, A. Qaddouri, M. Tanguay, J. Côté, A Fractional Steps Method for the Numerical Solution of the Shallow Water Equations, International Workshop on Solution of Partial Differential Equations, The Fields Institute, Toronto, August 2002], where the Riemann invariants of the equations are interpolated at each time step along the characteristics using a cubic spline interpolation. The potential vorticity, which develops steep gradients and evolve into thin filaments during the evolution, is nicely calculated at every time-step from the solution of the code. The method is efficient and has lower numerical diffusion than other methods, since it evolves the equations without the iterative steps involved in the multi-dimensional interpolation problem, and without the iteration associated with the intermediate step of solving a Helmholtz equation, usually associated with other methods like the semi-Lagrangian method. The absence of iterative steps in the present technique makes it very suitable for problems in which small time steps and grid sizes are required, as for instance in the present problem where steepness of the gradients and small scale structures are the main features of the potential vorticity, and more generally for problems of regional climate modeling. The simplicity of the method makes it very suitable for parallel computer.