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Shallow water equations: split-form, entropy stable, well-balanced, and positivity preserving numerical methods

  • Hendrik RanochaEmail author
Original Paper

Abstract

For the first time, a general two-parameter family of entropy conservative numerical fluxes for the shallow water equations is developed and investigated. These are adapted to a varying bottom topography in a well-balanced way, i.e. preserving the lake-at-rest steady state. Furthermore, these fluxes are used to create entropy stable and well-balanced split-form semidiscretisations based on general summation-by-parts (SBP) operators, including Gauß nodes. Moreover, positivity preservation is ensured using the framework of Zhang and Shu (Proc R Soc Lond A Math Phys Eng Sci 467: 2752–2776, 2011). Therefore, the new two-parameter family of entropy conservative fluxes is enhanced by dissipation operators and investigated with respect to positivity preservation. Additionally, some known entropy stable and positive numerical fluxes are compared. Furthermore, finite volume subcells adapted to nodal SBP bases with diagonal mass matrix are used. Finally, numerical tests of the proposed schemes are performed and some conclusions are presented.

Keywords

Skew-symmetric shallow water equations Summation-by-parts Split-form Entropy stability Well-balancedness Positivity preservation 

Mathematics Subject Classification

65M70 65M60 65M06 65M12 

Notes

Acknowledgements

The author would like to thank the anonymous reviewer for some helpful comments, resulting in an improved presentation of this material.

References

  1. Audusse, E., Bouchut, F., Bristeau, M.O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25(6), 2050–2065 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Audusse, E., Chalons, C., Ung, P.: A simple well-balanced and positive numerical scheme for the shallow-water system. Commun. Math. Sci. 13(5), 1317–1332 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Barth, T.J.: Numerical methods for gasdynamic systems on unstructured meshes. In: Ohlberger, M., Rohde, C., Kröner, D. (eds.) An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, pp. 195–285. Springer, Berlin (1999)CrossRefGoogle Scholar
  4. Berthon, C., Chalons, C.: A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations. Math. Comput. 85(299), 1281–1307 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. arXiv:1411.1607 (2014)
  6. Bouchut, F.: Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94(4), 623–672 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bouchut, F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-balanced Schemes for Sources. Springer Science & Business Media, New York (2004)CrossRefzbMATHGoogle Scholar
  8. Dafermos, C.M.: Hyperbolic conservation laws in continuum physics. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  9. Delestre, O., Cordier, S., Darboux, F., James, F.: A limitation of the hydrostatic reconstruction technique for shallow water equations. C. R. Math. 350(13), 677–681 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Delestre, O., Lucas, C., Ksinant, P.A., Darboux, F., Laguerre, C., Vo, T.N., James, F., Cordier, S., et al.: SWASHES: a compilation of shallow water analytic solutions for hydraulic and environmental studies. Int. J. Numer. Methods Fluids 72(3), 269–300 (2013) (used corrected version from arXiv). arXiv:1110.0288v7
  11. Dumbser, M., Zanotti, O., Loubère, R., Diot, S.: A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws. J. Comput. Phys. 278, 47–75 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Duran, A., Marche, F.: Recent advances on the discontinuous Galerkin method for shallow water equations with topography source terms. Comput. Fluids 101, 88–104 (2014)MathSciNetCrossRefGoogle Scholar
  13. Einfeldt, B.: On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal. 25(2), 294–318 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Einfeldt, B., Munz, C.D., Roe, P.L., Sjögreen, B.: On Godunov-type methods near low densities. J. Comput. Phys. 92(2), 273–295 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Fernández, D.C.D.R., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95, 171–196 (2014)MathSciNetCrossRefGoogle Scholar
  16. Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. Technical Report NASA/TM-2013-217971, NASA, NASA Langley Research Center, Hampton VA 23681-2199, United States (2013)Google Scholar
  17. Fisher, T.C., Carpenter, M.H., Nordström, J., Yamaleev, N.K., Swanson, C.: Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: theory and boundary conditions. J. Comput. Phys. 234, 353–375 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Fjordholm, U.S., Mishra, S., Tadmor, E.: Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys. 230(14), 5587–5609 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Frid, H.: Maps of convex sets and invariant regions for finite difference systems of conservation laws. Arch. Ration. Mech. Anal. 160(3), 245–269 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Frid, H.: Correction to “maps of convex sets and invariant regions for finite difference systems of conservation laws”. Arch. Ration. Mech. Anal. 171(2), 297–299 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Gassner, G.J.: A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35(3), A1233–A1253 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Gassner, G.J., Winters, A.R., Kopriva, D.A.: Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations (2016a). arXiv:1604.06618
  23. Gassner, G.J., Winters, A.R., Kopriva, D.A.: A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations. Appl. Math. Comput. 272, 291–308 (2016b)Google Scholar
  24. Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67(221), 73–85 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Gottlieb, S., Ketcheson, D.I., Shu, C.W.: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific, Hackensack (2011)CrossRefzbMATHGoogle Scholar
  26. Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev 25(1), 35–61 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Holden, H., Risebro, N.H.: Front Tracking for Hyperbolic Conservation Laws, vol. 152. Springer, Berlin (2002)zbMATHGoogle Scholar
  28. Huerta, A., Casoni, E., Peraire, J.: A simple shock-capturing technique for high-order discontinuous Galerkin methods. Int. J. Numer. Methods Fluids 69(10), 1614–1632 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Liu, X.D., Osher, S.: Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes I. SIAM J. Numer. Anal. 33(2), 760–779 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Meister, A., Ortleb, S.: A positivity preserving and well-balanced DG scheme using finite volume subcells in almost dry regions. Appl. Math. Comput. 272, 259–273 (2016)MathSciNetGoogle Scholar
  31. Ortleb S (2016) Kinetic energy preserving DG schemes based on summation-by-parts operators on interior node distributions. Talk presented at the joint annual meeting of DMV and GAMMGoogle Scholar
  32. Perthame, B., Simeoni, C.: A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38(4), 201–231 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Ranocha, H., Öffner, P., Sonar, T.: Extended skew-symmetric form for summation-by-parts operators (2015). Submitted. arXiv:1511.08408
  34. Ranocha, H., Öffner, P., Sonar, T.: Summation-by-parts operators for correction procedure via reconstruction. J. Comput. Phys. 311, 299–328 (2016). doi: 10.1016/j.jcp.2016.02.009
  35. Sonntag, M., Munz, C.D.: Shock capturing for discontinuous galerkin methods using finite volume subcells. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds.) Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, Springer Proceedings in Mathematics & Statistics, vol. 78, pp. 945–953. Springer International Publishing, Berlin (2014)Google Scholar
  36. Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  37. SymPy Development Team (2016) SymPy: Python library for symbolic mathematics. http://www.sympy.org
  38. Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49(179), 91–103 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  39. Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  40. Wintermeyer, N., Winters, A.R., Gassner, G.J., Kopriva, D.A. (2015) An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. arXiv:1509.07096v1
  41. Wintermeyer, N., Winters, A.R., Gassner, G.J., Kopriva, D.A.: An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry (2016). arXiv:1509.07096v2
  42. Xing, Y., Shu, C.W.: A survey of high order schemes for the shallow water equations. J. Math. Study 47(221–249), 56 (2014)MathSciNetGoogle Scholar
  43. Xing, Y., Zhang, X., Shu, C.W.: Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33(12), 1476–1493 (2010)CrossRefGoogle Scholar
  44. Zhang, X., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments. Proc. R. Soc. A Math. Phy. 467(2134), 2752–2776 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.TU BraunschweigBrunswickGermany

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