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Unconditionally Stable Splitting Methods for the Shallow Water Equations

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Abstract

The front-tracking method for hyperbolic conservation laws is combined with operator splitting to study the shallow water equations. Furthermore, the method includes adaptive grid refinement in multidimensions and shock tracking in one dimension. The front-tracking method is unconditionally stable, but for practical computations feasible CFL numbers are moderately above unity (typically between 1 and 5). The method resolves shocks sharply and is highly efficient. The numerical technique is applied to four test cases, the first being an expanding bore with rotational symmetry. The second problem addresses the question of describing the time development of two constant water levels separated by a dam that breaks instantaneously. The third problem compares the front-tracking method with an explicit analytic solution of water waves rotating over a parabolic bottom profile. Finally, we study flow over an obstacle in one dimension.

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REFERENCES

  1. F. Alcrudo and P. Garcia-Navarro, A high-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equations, Internat. J. Numer. Methods Fluids, 16 (1993), pp. 489–505.

    Google Scholar 

  2. D. Ambrosi, Approximation of shallow water equations by Roe's Riemann solver, Internat. J. Numer. Methods Fluids, 20 (1995), pp. 157–168.

    Google Scholar 

  3. A. Bermùdez, A. Dervieux, J. A. Désidéri, and M. E. Vázquez, Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Comput. Methods Appl. Mech. Engrg., 155 (1998), pp. 49–72.

    Google Scholar 

  4. A. Bressan, G. Crasta, and B. Piccoli, Well-posedness of the Cauchy problem for n × n systems of conservation laws, Mem. Amer. Math. Soc., to appear

  5. A. Bressan and P. LeFloch, Uniqueness of weak solutions to systems of conservation laws, Arch. Rational Mech. Anal., 140:4 (1997), pp. 301–317.

    Google Scholar 

  6. R. J. Fennema and M. H. Chaudhry, Explicit methods for 2-D transient free-surface flows, J. Hydraul. Eng., 116 (1990), pp. 1013–1034.

    Google Scholar 

  7. A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, New York, 1982.

    Google Scholar 

  8. H. Holden, K.-A. Lie, and N. H. Risebro, A front tracking method for conservation laws in two dimensions, J. Comput. Phys., 150 (1999), pp. 76–96.

    Google Scholar 

  9. H. Holden and N. H. Risebro, A method of fractional steps for scalar conservation laws without the CFL condition, Math. Comp., 60(201) (1993), pp. 221–232.

    Google Scholar 

  10. H. Holden and N. H. Risebro, Front Tracking for Conservation Laws, Lecture Notes, Department of Mathematical Sciences, Norwegian University of Science and Technology.

  11. G.-S. Jiang and E. Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 19:6 (1998), pp. 1892–1917.

    Google Scholar 

  12. J. O. Langseth, On an implementation of a front tracking method for hyperbolic conservation laws, Advances in Engineering Software, 26:1 (1996), pp. 45–63.

    Google Scholar 

  13. J. O. Langseth, N. H. Risebro, and A. Tveito, A conservative front tracking scheme for 1D hyperbolic conservation laws, in Nonlinear Hyperbolic Problems: Theoretical, Applied, and Computational Aspects (Taormina, 1992), A. Donato and F. Oliveri, eds., Vol. 43 of Notes Numer. Fluid Mech., Vieweg, Braunschweig, 1993, pp. 385–392.

    Google Scholar 

  14. R. J. LeVeque, Large time step shock-capturing techniques for scalar conservation laws, SIAM J. Numer. Anal., 22 (1985), pp. 1051–1073.

    Google Scholar 

  15. R. J. LeVeque, Clawpack software.

  16. R. J. LeVeque, Clawpack User Notes, Applied Mathematics, Box 352420, University of Washington, Seattle, WA 98195–2420, Nov. 1995.

    Google Scholar 

  17. R. J. LeVeque, Wave propagation algorithms for multidimensional hyperbolic systems, J. Comput. Phys. 131 (1997), pp. 327–353.

    Google Scholar 

  18. R. J. LeVeque and K. M. Shyue, One-dimensional front tracking based on high resolution wave propagation methods, SIAM J. Sci. Comput., 16:2 (1995), pp. 348–377.

    Google Scholar 

  19. K.-A. Lie, Front Tracking and Operator Splitting for Convection Dominated Problems. Dr. ing. thesis, NTNU Trondheim, 1998.

  20. K.-A. Lie, V. Haugse, and K. H. Karlsen, Dimensional splitting with front tracking and adaptive grid refinement, Numer. Methods Partial Differential Equations, 14:5 (1998), pp. 627–648.

    Google Scholar 

  21. R. Liska, L. Margolin, and B. Wendroff, Nonhydrostatic two-layer models of incompressible flow, Comput. Math. Appl., 29 (1995), pp. 25–37.

    Google Scholar 

  22. R. Liska and B. Wendroff, Composite schemes for conservation laws, SIAM J. Numer. Anal., 35 (1998), pp. 2250–2271.

    Google Scholar 

  23. R. Liska and B. Wendroff, 2D shallow water equations by composite schemes, J. Num. Meth. Fluids, to appear.

  24. R. Liska and B. Wendroff, Analysis and computation with stratified fluid models, J. Comput. Phys., 137:1 (1997), pp. 212–244.

    Google Scholar 

  25. T.-P. Liu and J. Smoller, On the vacuum state for isentropic gas dynamics, Adv. Pure Appl. Math., 1 (1980), pp. 345–359.

    Google Scholar 

  26. A. T. Morel, A Genuinely Multidimensional High-Resolution Scheme for the Shallow Water Equations, PhD Thesis, Swiss Federal Institute of Technology, Zürich, 1997.

    Google Scholar 

  27. A. N. Nazarov, Mathematical modeling of a snow-powder avalanche in the framework of the equations of two-layer shallow water, Fluid Dynamics, 26 (1991), pp. 70–75.

    Google Scholar 

  28. N. H. Risebro, A front-tracking alternative to the random choice method, Proc. Amer. Math. Soc, 117:4 (1993), pp. 1125–1129.

    Google Scholar 

  29. N. H. Risebro and A. Tveito, Front tracking applied to a nonstrictly hyperbolic system of conservation laws, SIAM J. Sci. Stat. Comput., 12:6 (1991), pp. 1401–1419.

    Google Scholar 

  30. N. H. Risebro and A. Tveito, A front tracking method for conservation laws in one dimension, J. Comput. Phys., 101:1 (1992), pp. 130–139.

    Google Scholar 

  31. B. K. Swartz and B. Wendroff, AZTEC: a front tracking code based on Godunov's method, Appl. Numer. Math., 2:3–5 (1986), pp. 385–397.

    Google Scholar 

  32. W. C. Thacker, Some exact solutions to the nonlinear shallow-water wave equations, J. Fluid Mech., 107 (1981), pp. 499–508.

    Google Scholar 

  33. E. F. Toro, Riemann problems and the WAF method for solving the two-dimensional shallow water equations, Philos. Trans. Roy. Soc. London Ser. A., 338 (1992), pp. 43–68.

    Google Scholar 

  34. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Mechanics, Springer-Verlag, Berlin, 1997.

    Google Scholar 

  35. E. F. Toro and S. J. Billett, A unified Riemann-problem-based extension of the Warming-Beam and Lax-Wendroff schemes, IMA J. Numer. Anal., 17 (1997), pp. 61–102.

    Google Scholar 

  36. C. B. Vreugdenhil, Two-layer shallow-water flow in two dimensions, a numerical study, J. Comput. Phys., 33 (1979), pp. 169–184.

    Google Scholar 

  37. B. Wendroff, An analysis of front tracking for chromatography, Acta Appl. Math., 30 (1993), pp. 265–285.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway. email: holdahl@math.ntnu.no

    Runar Holdahl & Helge Holden

  2. Department of Informatics, University of Oslo, P.O. Box 1080 Blindern, NO-0316, Oslo, Norway. email: kalie@ifi.uio.no

    Knut-Andreas Lie

Authors
  1. Runar Holdahl
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  2. Helge Holden
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  3. Knut-Andreas Lie
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Holdahl, R., Holden, H. & Lie, KA. Unconditionally Stable Splitting Methods for the Shallow Water Equations. BIT Numerical Mathematics 39, 451–472 (1999). https://doi.org/10.1023/A:1022366502335

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