Paper title: Fast Fourier Transform Algorithms for the Nonlinear Shallow Water Equations DOI: https://doi.org/10.4316/JACSM.202002006 Published in: Issue 2, (Vol. 14) / 2020Download Publishing date: 2020-07-14 Pages: 36-41 Author(s): PUTHENVEETTIL Saumya, MANDLI Kyle Abstract. 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. Keywords: Nonlinear Shallow Water Equations, Partial Differential Equations, Hyperbolic, Spectral, Pseudo-Spectral, Numerical Computation, Tsunami, Storm Surge, Fast Fourier Transform, Chebyshev Differentiation, Integrating Factor, Fluid Dynamics References: 1. Y. Liu, Y. Shi, D. Yuen, E. Sevre, X. Yuan, H. Xing, Comparison of linear and nonlinear shallow wave water equations applied to tsunami waves over the China Sea, Acta Geotechnica.,4 (2009), pp. 129-137. 2. K. Mandli, Finite Volume Methods for the Multilayer Shallow Water Equations with Applications to Storm Surges, Ph.D. thesis, University of Washington, Seattle, WA, 2011. 3. U. Fjordholm, S. Mishra, Vorticity Preserving Finite Volume Schemes for the Shallow Water Equations, SIAM Journal of Scientific Computing, 33 (2011), pp. 588-611. 4. P. Merilees, The Pseudospectral Approximation Applied to the Shallow Water Equations on a Sphere, Atmosphere, 11(1973), pp. 13-20. 5. J. Li, H. Li, Numerical Simulation of Water Wave Based on Chebyshev Spectral Method, International Journal of Applied and Natural Sciences, 8 (2019), pp. 153-160. 6. D. Dutykh, A Brief Introduction to Pseudo-Spectral Methods: Application to Diffusion Problems, arXiv: 1606.05432v2 math.NA, 2019. 7. L. Trefethen, Spectral Methods in Matlab, Society of Industrial and Applied Mathematics, Philadelphia, PA, 2000. 8. M. Brocchini, N. Dodd, Nonlinear Shallow Water Equation Modeling for Coastal Engineering, Journal of Waterway Port Coastal and Ocean Engineering, 134 (2008), pp. 104-120. 9. H. Altaie, P. Dreyfuss, Numerical Solutions for 2D Depth-Averaged Shallow Water Equations, International Mathematical Forum, 13 (2018), pp. 79-90. Back to the journal content