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

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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.

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References

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Chapter  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. arXiv:1411.1607 (2014)

  • Bouchut, F.: Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94(4), 623–672 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Book  MATH  Google Scholar 

  • Dafermos, C.M.: Hyperbolic conservation laws in continuum physics. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • Einfeldt, B.: On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal. 25(2), 294–318 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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

  • 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)

  • Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67(221), 73–85 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Gottlieb, S., Ketcheson, D.I., Shu, C.W.: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific, Hackensack (2011)

    Book  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Holden, H., Risebro, N.H.: Front Tracking for Hyperbolic Conservation Laws, vol. 152. Springer, Berlin (2002)

    MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  Google Scholar 

  • 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 GAMM

  • Perthame, B., Simeoni, C.: A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38(4), 201–231 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Ranocha, H., Öffner, P., Sonar, T.: Extended skew-symmetric form for summation-by-parts operators (2015). Submitted. arXiv:1511.08408

  • 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

  • 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)

  • Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • SymPy Development Team (2016) SymPy: Python library for symbolic mathematics. http://www.sympy.org

  • Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49(179), 91–103 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • 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

  • 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

  • Xing, Y., Shu, C.W.: A survey of high order schemes for the shallow water equations. J. Math. Study 47(221–249), 56 (2014)

    MathSciNet  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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  1. TU Braunschweig, Pockelsstraße 14, 38106, Brunswick, Germany

    Hendrik Ranocha

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  1. Hendrik Ranocha
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Ranocha, H. Shallow water equations: split-form, entropy stable, well-balanced, and positivity preserving numerical methods. Int J Geomath 8, 85–133 (2017). https://doi.org/10.1007/s13137-016-0089-9

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