Fast Fourier Transform Algorithms for the Nonlinear Shallow Water Equations

Discretization of the two dimensional nonlinear shallow water equations is done using pseudo-spectral methods with a translating vortex test case. First, the equations are modeled using Chebyshev differentiation with Fast Fourier Transform. The test case is an exact solution of the shallow water equations which is a traveling vortex defined by constants of any value. The equations are integrated using Euler and Leapfrog time stepping methods and convergence of the solutions is computed using the infinite norm. Second, this solution is compared to discretization of the equations using the method of integrating factors and fourth order Runge-Kutta for integration. The method of integrating factors uses the zero-padding technique for anti-aliasing of the nonlinear terms. The solution’s streamlines are also compared to that of the true solution to test for accuracy and determine the direction of the flow. A second test case is a perturbation of fluid for specific times. Results at .02 seconds show that the Chebyshev differentiation method can produce first and third order convergence using the Euler and Leapfrog methods and the Integrating factor method is able to construct a solution after de-aliasing.