Splitting Modified Donor-Cell Schemes for Spectral Action Balance Equation
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 propagation velocity terms are discretized using central
differences, stability problems occur when the grid spacing is chosen
too coarse. In this paper, we introduce the splitting modified donorcell
scheme for avoiding stability problems and prove that it is
consistent to the modified donor-cell scheme with same accuracy. The
splitting modified donor-cell scheme was adopted to split the wave
spectral action balance equation into four one-dimensional 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-cores computer.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:53702", author = "Tanapat Brikshavana and Anirut Luadsong", title = "Splitting Modified Donor-Cell Schemes for Spectral Action Balance Equation", 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 propagation velocity terms are discretized using central
differences, stability problems occur when the grid spacing is chosen
too coarse. In this paper, we introduce the splitting modified donorcell
scheme for avoiding stability problems and prove that it is
consistent to the modified donor-cell scheme with same accuracy. The
splitting modified donor-cell scheme was adopted to split the wave
spectral action balance equation into four one-dimensional 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-cores computer.", keywords = "donor-cell scheme, parallel algorithm, spectral action balance equation, splitting method.", volume = "4", number = "7", pages = "856-9", }