Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method

ABSTRACT

The two-dimensional nonlinear shallow water equations in the presence of Coriolis force and bottom topography are solved numerically using the fractional steps method. The fractional steps method consists of splitting the multi-dimensional matrix inversion problem into an equivalent one dimensional problem which is successively integrated in every direction along the characteristics using the Riemann invariant associated with the cubic spline interpolation. The height and the velocity field of the shallow water equations over irregular bottom are discretized on a fixed Eulerian grid and time-stepped using the fractional steps method. Effects of the Coriolis force and the bottom topography for particular initial flows on the velocity components and the free surface elevation have been studied and the results are plotted.

The two-dimensional nonlinear shallow water equations in the presence of Coriolis force and bottom topography are solved numerically using the fractional steps method. The fractional steps method consists of splitting the multi-dimensional matrix inversion problem into an equivalent one dimensional problem which is successively integrated in every direction along the characteristics using the Riemann invariant associated with the cubic spline interpolation. The height and the velocity field of the shallow water equations over irregular bottom are discretized on a fixed Eulerian grid and time-stepped using the fractional steps method. Effects of the Coriolis force and the bottom topography for particular initial flows on the velocity components and the free surface elevation have been studied and the results are plotted.

KEYWORDS

Shallow Water Equations, Fractional Steps Method, Riemann Invariants, Bottom Topography, Cubic Spline Interpolation

Shallow Water Equations, Fractional Steps Method, Riemann Invariants, Bottom Topography, Cubic Spline Interpolation

Cite this paper

nullH. Hassan, K. Ramadan and S. Hanna, "Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method,"*Applied Mathematics*, Vol. 1 No. 2, 2010, pp. 104-117. doi: 10.4236/am.2010.12014.

nullH. Hassan, K. Ramadan and S. Hanna, "Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method,"

References

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[2] J. Pedlosky, “Geophysical Fluid Dynamics,” Springer, New York, 1987.

[3] M. Lukácová-Medvid'ová, S. Noelle and M. Kraft, “Well- Balanced Finite Volume Evolution Galerkin Methods for the Shallow Water Equations,” Journal of Computational Physics, Vol. 221, No. 1, 2007, pp. 122-147.

[4] T. Gallou?t, J. M. Hérard and N. Seguin, “Some Approximate Godunov Schemes to Compute Shallow-Water Equations with Topography,” Computers & Fluids, Vol. 32, No. 4, 2003, pp. 479-513.

[5] P. J. Dellar and R. Salmon, “Shallow Water Equations with a Complete Coriolis Force and Topography,” Physics of Fluids, Vol. 17, No. 10, 2005, pp. 106601-106619.

[6] K. V. Karelsky, V. V.Papkov, A. S. Petrosyan and D. V. Tsygankov, “Particular Solution of the Shallow-Water Equations over a Non-Flat Surface,” Physics Letters A, Vol. 271, No. 5-6, 2000, pp. 341-348.

[7] D. L. George, “Augmented Riemann Solvers for the Shallow Water Equations over Variable Topography with Steady States and Inundation,” Journal of Computational Physics, Vol. 227, No. 6, 2008, pp. 3089-3113.

[8] M. Shoucri, “The Application of a Fractional Steps Method for the Numerical Solution of the Shallow Water Equations,” Computer Physics Communications, Vol. 164, No. 1-3, 2004, pp. 396-401.

[9] A. Stainiforth and C. Temperton, “Semi-Implicit Semi- Lagrangian Integration Scheme for a Baratropic Finite- Element Regional Model,” Monthly Weather Review, Vol. 114, No. 11, 1986, pp. 2078-2090.

[10] M. B. Abd-el-Malek and M. H. Helal, “Application of the Fractional Steps Method for the Numerical Solution of the Two-Dimensional Modeling of the Lake Mariut,” Applied Mathematical Modeling, Vol. 33, No. 2, 2009, pp. 822-834.

[11] M. Shoucri, “Numerical Solution of the Shallow Water Equations with a Fractional Step Method,” Computer Physics Communications, Vol. 176, No. 1, 2007, pp. 23-32.

[12] I. Yohsuke, A. Takayuki and M. Shoucri, “Comparison of Efficient Explicit Schemes for Shallow-Water Equations- Characteristics-Based Fractional-Step Method and Multimoment Eulerian Scheme,” Journal of Applied Meteorology and Climatology, Vol. 46, No. 3, 2007, pp. 388-395.

[13] A. Takayuki, “Interpolated Differential Operator (IDO) Scheme for Solving Partial Differential Equations,” Computer Physics Communications, Vol. 102, No. 1-3, 1997, pp. 132-146.

[14] C. Temperton and A. Stainiforth, “An Efficient Two- Time-Level Semi-Lagrangian Semi-Implicit Integration Scheme,” Quarterly Journal of the Royal Meteorological Society, Vol. 113, No. 477, 1987, pp. 1025-1039.

[15] E. Audusse, F. Bouchut, M. Bristeau, R. Klein and B. Per- thame, “A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows,” SIAM Journal on Scientific Computing, Vol. 25, No. 6, 2004, pp. 2050-2065.

[16] M. E. Talibi and M. H. Tber, “On a Problem of Shallow Water Type,” Electronic Journal of Differential Equations, Vol. 11, 2004, pp. 109-116.

[17] V. R. Ambati and O. Bokhove, “Space-Time Discontinuous Galerkin Discretization of Rotating Shallow Water Equations,” Journal of Computational Physics, Vol. 225, No. 2, 2007, pp. 1233-1261.

[18] N. N. Yanenko, “The Method of Fractional Steps, the Solution of Problems of Mathematical Physics in Several Variables,” Springer-Verlag, Berlin, 1971.

[19] D. R. Durran, “Numerical Methods for Wave Equations in Geophysics Fluid Dynamics,” Springer-Verlag, New York, 1999.

[20] G. Strang, “On the Construction and Comparison of Finite Difference Schemes,” Society for Industrial and Applied Mathematics, Journal for Numerical Analysis, Vol. 5, No. 3, 1968, pp. 506-517.

[21] R. L. Burden and J. D. Faires, “Numerical Analysis,” PWS Publishing Company, Boston, 1993.

[22] S. Noelle, N. Pankratz, G. Puppo and J. R. Natvig, “Well-Balanced Finite Volume Schemes of Arbitrary Order of Accuracy for Shallow Water Flows,” Journal of Computational Physics, Vol. 213, No. 2, 2006, pp. 474-499.

[23] S. Noelle, Y. Xing and C. Shu, “High-Order Well-Balanced Finite Volume WENO Schemes for Shallow Water Equation with Moving Water,” Journal of Computational Physics, Vol. 226, No. 1, 2007, pp. 29-58.

[24] R. LeVeque, “Finite Volume Methods for Hyperbolic Pro- blems,” Cambridge University Press, Cambridge, 2004.

[25] Y. Z. Boutros, M. B. Abd-el-Malek, N. A. Badran and H. S. Hassan, “Lie-Group Method for Unsteady Flows in a Semi-infinite Expanding or Contracting Pipe with Injection or Suction through a Porous Wall,” Journal of Computational and Applied Mathematics, Vol. 197, No. 2, 2006, pp. 465-494.

[26] Y. Z. Boutros, M. B. Abd-el-Malek, N. A. Badran and H. S. Hassan, “Lie-Group Method Solution for Two-dimen- sional Viscous Flow between Slowly Expanding or Contracting Walls with Weak Permeability,” Applied Mathematical Modelling, Vol. 31, No. 6, 2007, pp. 1092-1108.

[27] Y. Z. Boutros, M. B. Abd-el-Malek, N. A. Badran and H. S. Hassan, “Lie-Group Method of Solution for Steady Two-Dimensional Boundary-Layer Stagnation-Point Flow Towards a Heated Stretching Sheet Placed in a Porous Medium,” Meccanica, Vol. 41, No. 6, 2007, pp. 681-691.

[28] M. B. Abd-el-Malek, N. A. Badran and H. S. Hassan, “Lie-Group Method for Predicting Water Content for Immiscible Flow of Two Fluids in a Porous Medium,” Applied Mathematical Sciences, Vol. 1, No. 24, 2007, pp. 1169-1180.

[29] M. B. Abd-el-Malek and H. S. Hassan, “Symmetry Analysis for Solving Problem of Rivlin-Ericksen Fluid of Second Grade Subject to Suction,” submitted for publication.

[1] D. G. Dritschel, L. M. Polvani and A. R. Mohebalhojeh, “The Contour-Advective Semi-Lagrangian Algorithm for the Shallow Water Equations,” Monthly Weather Review, Vol. 127, No. 7, 1999, pp. 1551-1564.

[2] J. Pedlosky, “Geophysical Fluid Dynamics,” Springer, New York, 1987.

[3] M. Lukácová-Medvid'ová, S. Noelle and M. Kraft, “Well- Balanced Finite Volume Evolution Galerkin Methods for the Shallow Water Equations,” Journal of Computational Physics, Vol. 221, No. 1, 2007, pp. 122-147.

[4] T. Gallou?t, J. M. Hérard and N. Seguin, “Some Approximate Godunov Schemes to Compute Shallow-Water Equations with Topography,” Computers & Fluids, Vol. 32, No. 4, 2003, pp. 479-513.

[5] P. J. Dellar and R. Salmon, “Shallow Water Equations with a Complete Coriolis Force and Topography,” Physics of Fluids, Vol. 17, No. 10, 2005, pp. 106601-106619.

[6] K. V. Karelsky, V. V.Papkov, A. S. Petrosyan and D. V. Tsygankov, “Particular Solution of the Shallow-Water Equations over a Non-Flat Surface,” Physics Letters A, Vol. 271, No. 5-6, 2000, pp. 341-348.

[7] D. L. George, “Augmented Riemann Solvers for the Shallow Water Equations over Variable Topography with Steady States and Inundation,” Journal of Computational Physics, Vol. 227, No. 6, 2008, pp. 3089-3113.

[8] M. Shoucri, “The Application of a Fractional Steps Method for the Numerical Solution of the Shallow Water Equations,” Computer Physics Communications, Vol. 164, No. 1-3, 2004, pp. 396-401.

[9] A. Stainiforth and C. Temperton, “Semi-Implicit Semi- Lagrangian Integration Scheme for a Baratropic Finite- Element Regional Model,” Monthly Weather Review, Vol. 114, No. 11, 1986, pp. 2078-2090.

[10] M. B. Abd-el-Malek and M. H. Helal, “Application of the Fractional Steps Method for the Numerical Solution of the Two-Dimensional Modeling of the Lake Mariut,” Applied Mathematical Modeling, Vol. 33, No. 2, 2009, pp. 822-834.

[11] M. Shoucri, “Numerical Solution of the Shallow Water Equations with a Fractional Step Method,” Computer Physics Communications, Vol. 176, No. 1, 2007, pp. 23-32.

[12] I. Yohsuke, A. Takayuki and M. Shoucri, “Comparison of Efficient Explicit Schemes for Shallow-Water Equations- Characteristics-Based Fractional-Step Method and Multimoment Eulerian Scheme,” Journal of Applied Meteorology and Climatology, Vol. 46, No. 3, 2007, pp. 388-395.

[13] A. Takayuki, “Interpolated Differential Operator (IDO) Scheme for Solving Partial Differential Equations,” Computer Physics Communications, Vol. 102, No. 1-3, 1997, pp. 132-146.

[14] C. Temperton and A. Stainiforth, “An Efficient Two- Time-Level Semi-Lagrangian Semi-Implicit Integration Scheme,” Quarterly Journal of the Royal Meteorological Society, Vol. 113, No. 477, 1987, pp. 1025-1039.

[15] E. Audusse, F. Bouchut, M. Bristeau, R. Klein and B. Per- thame, “A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows,” SIAM Journal on Scientific Computing, Vol. 25, No. 6, 2004, pp. 2050-2065.

[16] M. E. Talibi and M. H. Tber, “On a Problem of Shallow Water Type,” Electronic Journal of Differential Equations, Vol. 11, 2004, pp. 109-116.

[17] V. R. Ambati and O. Bokhove, “Space-Time Discontinuous Galerkin Discretization of Rotating Shallow Water Equations,” Journal of Computational Physics, Vol. 225, No. 2, 2007, pp. 1233-1261.

[18] N. N. Yanenko, “The Method of Fractional Steps, the Solution of Problems of Mathematical Physics in Several Variables,” Springer-Verlag, Berlin, 1971.

[19] D. R. Durran, “Numerical Methods for Wave Equations in Geophysics Fluid Dynamics,” Springer-Verlag, New York, 1999.

[20] G. Strang, “On the Construction and Comparison of Finite Difference Schemes,” Society for Industrial and Applied Mathematics, Journal for Numerical Analysis, Vol. 5, No. 3, 1968, pp. 506-517.

[21] R. L. Burden and J. D. Faires, “Numerical Analysis,” PWS Publishing Company, Boston, 1993.

[22] S. Noelle, N. Pankratz, G. Puppo and J. R. Natvig, “Well-Balanced Finite Volume Schemes of Arbitrary Order of Accuracy for Shallow Water Flows,” Journal of Computational Physics, Vol. 213, No. 2, 2006, pp. 474-499.

[23] S. Noelle, Y. Xing and C. Shu, “High-Order Well-Balanced Finite Volume WENO Schemes for Shallow Water Equation with Moving Water,” Journal of Computational Physics, Vol. 226, No. 1, 2007, pp. 29-58.

[24] R. LeVeque, “Finite Volume Methods for Hyperbolic Pro- blems,” Cambridge University Press, Cambridge, 2004.

[25] Y. Z. Boutros, M. B. Abd-el-Malek, N. A. Badran and H. S. Hassan, “Lie-Group Method for Unsteady Flows in a Semi-infinite Expanding or Contracting Pipe with Injection or Suction through a Porous Wall,” Journal of Computational and Applied Mathematics, Vol. 197, No. 2, 2006, pp. 465-494.

[26] Y. Z. Boutros, M. B. Abd-el-Malek, N. A. Badran and H. S. Hassan, “Lie-Group Method Solution for Two-dimen- sional Viscous Flow between Slowly Expanding or Contracting Walls with Weak Permeability,” Applied Mathematical Modelling, Vol. 31, No. 6, 2007, pp. 1092-1108.

[27] Y. Z. Boutros, M. B. Abd-el-Malek, N. A. Badran and H. S. Hassan, “Lie-Group Method of Solution for Steady Two-Dimensional Boundary-Layer Stagnation-Point Flow Towards a Heated Stretching Sheet Placed in a Porous Medium,” Meccanica, Vol. 41, No. 6, 2007, pp. 681-691.

[28] M. B. Abd-el-Malek, N. A. Badran and H. S. Hassan, “Lie-Group Method for Predicting Water Content for Immiscible Flow of Two Fluids in a Porous Medium,” Applied Mathematical Sciences, Vol. 1, No. 24, 2007, pp. 1169-1180.

[29] M. B. Abd-el-Malek and H. S. Hassan, “Symmetry Analysis for Solving Problem of Rivlin-Ericksen Fluid of Second Grade Subject to Suction,” submitted for publication.