A NEW VARIANT OF RUSANOV SCHEME: β-RUSANOV
FOR NUMERICAL RESOLUTION OF SHALLOW
WATER FLOWS
Maman Yarodji Abdoul Kader1, Badé Rabé2, Saley Bisso3
1,2,3Department of Mathematics and Computer Science
Faculty of Science and Technology
Abdou Moumouni University, Niamey, NIGER

Abstract

In this article, we are interested in the numerical resolution of the shallow water equation. We present a new modified version of the Rusanov scheme, called β-Rusanov. A mathematical analysis of this new scheme is done. For the approximation of the source term we use the hydrostatic reconstruction method. Some, numerical tests are performed to show the efficiency of our new scheme and compare this one with the classical Rusanov scheme.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 35
Issue: 4
Year: 2022

DOI: 10.12732/ijam.v35i4.7

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References

  1. [1] W.K. Anderson, J.L. Thomas and B. van Leer, Comparison of finite volume flux vector splittings for the Euler equations, AIAA Journal, 24, No 9 (1986), 1453-1460.
  2. [2] E. Audusse, F. Bouchut, M. O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25, No 6 (2004), 2050-2065.
  3. [3] F. Benkhaldoun, I. Elmahi and M. Seaid, Well-balanced finite volume schemes for pollutant transport on unstructured meshes, J. of Comput. Phys., 226, No 1 (2006), 180-203.
  4. [4] T. Buffard, T. Gallouet, and J.M. Hérard, A sequel to a rough godunov scheme: application to real gases, Comput. Fluids, 29 (2000) 813–847.
  5. [5] O. Delestre, S. Cordier, F. James and F. Darbaux, Simulation of rain-water Overland-Flow, In: 12th International Conference on Hyperbolic Problem, College Park (2008), 537-546.
  6. [6] H. Derakhshan and M.Z. Bistkonj, A new two dimensional model for pollutant transport in ajichai river, J. of Hydraulic Structures, 1, No 2 (2013), 44-54.
  7. [7] A. Duran, Q. Liang and F. Marche, On the well-balanced numerical discretization of shallow water equations on instructured meshes, J. of Computational Physics, 235 (2013), 565-585.
  8. [8] G. Edwige and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Collection Applied Mathematical Sciences, 118, Springer-Verlag, New York (1996).
  9. [9] J.M. Ghidaglia, Flux schemes for solving nonlinear systems of conservation laws, In: Proc. Meeting in Honour of P.L. Roe, Arcachon, France.
  10. [10] E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Ser. Applied Mathematical Sciences, 118, 2nd Ed., Springer.
  11. [11] S.K. Godunov, A difference method for numerical calculation of discontinious solutions of the equations of hydrodynamics, Math. Sb., 47 (89) (1959), 271-306.
  12. [12] A. Harten, P.D. Lax, and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review, 25 (1983), 35-61.
  13. [13] J.M. Hervouet, Hydrodynamics of Free Surface Flows Modelling with the Finite Element Method, John Wiley & Sons (2007).
  14. [14] M.Y.A. Kader, R. Badé and B. Saley, Study of the 1D Saint-Venant equations and application to the simulation of a flood problem, J. of Applied Mathematics and Physics, 8 (2020), 1193-1206.
  15. [15] I.M. Kulikov, I.G. Chernykh, A.F. Sapetina, S.V. Lomakin and A.V. Tutukov, A new Rusanov-Type solver with a local linear solution reconstruction for numerical modeling of white dwarf mergers by means massive parallel supercomputers, Lobachevskii J. of Mathematics, 41, No 8 (2020), 1485–1491.
  16. [16] P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), 217-237.
  17. [17] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge (2002).
  18. [18] K. Mohamed and M.A.E. Abdelrahman, The modified Rusanov Scheme for solving the ultra-relativistic Euler equations, European J. of Mechanics/ B Fluids, 90 (2021), 89-98.
  19. [19] K. Mohamed, and F. Benkhaldoun, The modified Rusanov scheme for shallow water equations with topography and two phase flows, Eur. Phys. J. Plus (2016), 131-207.
  20. [20] K. Mohamed, M. Seaid, and M. Zahri, A finite volume method for scalar conservation laws with stochastic time-space dependent flux functions, J. of Comput. and Appl. Math. 237 (2013), 614–632.
  21. [21] P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. of Comput. Phys., 43 (1981), 357-372.
  22. [22] V.V. Rusanov, The calculaion of the interaction of non-stationary shock waves and obstacles, Zh. Vych. Mat., 1, No 2 (1961), 267-279.
  23. [23] J.J. Stoker, Water Waves: The Mathematical Theory with Applications, Interscience, New York (1957).
  24. [24] E.F. Toro, M. Spruce, and W. Speare, Restoration of the contact surface in the HLL Riemann solver, Shock Waves, 4 (1994), 25-34.