The Splitting Upwind Schemes for Spectral Action Balance Equation
Commenced in January 2007
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Edition: International
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The Splitting Upwind Schemes for Spectral Action Balance Equation

Authors: Anirut Luadsong, Nitima Aschariyaphotha

Abstract:

The spectral action balance equation is an equation that used to simulate short-crested wind-generated waves in shallow water areas such as coastal regions and inland waters. This equation consists of two spatial dimensions, wave direction, and wave frequency which can be solved by finite difference method. When this equation with dominating convection term are discretized using central differences, stability problems occur when the grid spacing is chosen too coarse. In this paper, we introduce the splitting upwind schemes for avoiding stability problems and prove that it is consistent to the upwind scheme with same accuracy. The splitting upwind schemes was adopted to split the wave spectral action balance equation into four onedimensional problems, which for each small problem obtains the independently tridiagonal linear systems. For each smaller system can be solved by direct or iterative methods at the same time which is very fast when performed by a multi-processor computer.

Keywords: upwind scheme, parallel algorithm, spectral action balance equation, splitting method.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1079652

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[1] Booij, N., Ris, R.C. and Holthuijsen, L.H., A third-generation wave model for coastal regions:1. Model description and validation, Journal of Geophysical Research, 104(1999), 7649-7666.
[2] Brikshavana, T. and Luadsong, A., Fractional-step Method for Spectral Action Balance Equation, Far East Journal of Mathematical Sciences (FJMS), Volume 37(2)2010, 193-207.
[3] Frochte, J. and Heinrichs, W., A splitting technique of higher order for the Navier-Stokes equations, Journal of Computational and Applied Mathematics, 228(2009), 373-390.
[4] Griebel, M., Dornseifer, T. and Neunhoeffer, T., Numerical Simulation in Fluid Dynamics: a practical introduction, Siam monographs on mathematical modeling and computation, 1997.
[5] Hirt, C, Nichols, B., & Romero, N., SOLA - A Numerical Solution Algorithm for Transient Fluid Flows. Technical report LA-5852, Los Alamos, NM: Los Alamos National Lab, 1975.
[6] Luadsong, A., Finite-Difference method for shape preserving spline interpolation, Suranaree University of Technology, ISBN 9745331813, 2002.
[7] Ris, R.C., Holthuijsen L.H. and Booij N., A third-generation wave model for coastal regions:2. Verification, Journal of Geophysical Research, 104(1999), 7667-7681.
[8] Tolman, H.L., A Third-Generation Model for Wind Waves on Slowly varying, Unsteady, and Inhomogeneous Depths and Currents, Journal of Physical Oceanography, 21(1991), 782-797.
[9] WAMDI Group, The WAM Model-A Third Generation Ocean Wave Prediction Model, Journal of Physical Oceanography, 18(1988), 1775- 1810.
[10] Yanenko, N.N., The method of fractional steps, the solution of problems of Mathematical Physics in several variables, Springer-Verlag New York Heidelberg Berlin, 1971.
[11] Yan, Y., Xu, F. and Mao, L., A new type numerical model for action balance equation in simulating nearshore waves, Chinese Science Bullettin, 46(2001), 1-6.