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Entropy Stable and Well-Balanced Discontinuous Galerkin Methods for the Nonlinear Shallow Water Equations

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Abstract

The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas, with extensive applications in ocean and hydraulic engineering. In this work, we propose entropy stable, well-balanced and positivity-preserving discontinuous Galerkin (DG) methods, under arbitrary choices of quadrature rules, for the SWEs with a non-flat bottom topography. In Chan (J Comput Phys 362:346–374, 2018), a SBP-like differentiation operator was introduced to construct the discretely entropy conservative DG methods. We extend this idea to the SWEs and establish an entropy stable scheme by adding additional dissipative terms. Careful approximation of the source term is included to ensure the well-balanced property of the resulting method. A simple positivity-preserving limiter, compatible with the entropy stable property, is included to guarantee the non-negative water heights during the computation. One- and two-dimensional numerical experiments are presented to demonstrate the performance of the proposed methods.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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, 2050–2065 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bale, D.S., LeVeque, R.J., Mitran, S., Rossmanith, J.A.: A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24, 955–978 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berthon, C., Marche, F.: A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes. SIAM J. Sci. Comput. 30, 2587–2612 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bermudez, A., Vazquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bokhove, O.: Flooding and drying in discontinuous Galerkin finite-element discretizations of shallow-water equations. Part 1: one dimension. J. Sci. Comput. 22, 47–82 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bollermann, A., Noelle, S., Lukácová-Medviová, M.: Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys. 10, 371–404 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bunya, S., Kubatko, E.J., Westerink, J.J., Dawson, C.: A wetting and drying treatment for the Runge–Kutta discontinuous Galerkin solution to the shallow water equations. Methods Appl. Mech. Eng. 198, 1548–1562 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carpenter, M., Fisher, T., Nielsen, E., Frankel, S.: Entropy stable spectral collocation schemes for the Navier–Stokes equations: discontinuous interfaces. SIAM J. Sci. Comput. 36(5), B835–B867 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chan, J.: On discretely entropy conservative and entropy stable discontinuous Galerkin methods. J. Comput. Phys. 362, 346–374 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chan, J.: On discretely entropy conservative and entropy stable discontinuous Galerkin methods. arXiv:1708.01243v4 [math.NA]

  11. Chen, T., Shu, C.-W.: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427–461 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, T., Shu, C.-W.: Review of entropy stable discontinuous Galerkin methods for systems of conservation laws on unstructured simplex meshes. CSIAM Trans. Appl. Math. (CSAM) (2020). https://doi.org/10.4208/csiam-am.2020-0003

    Article  Google Scholar 

  13. Cockburn, B., Karniadakis, G., Shu, C.-W.: The development of discontinuous Galerkin methods. In: Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, Part I: Overview, vol. 11, pp. 5–50. Springer, New York (2000)

    Google Scholar 

  14. Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)

    MathSciNet  MATH  Google Scholar 

  15. Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dawson, C., Proft, J.: Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations. Comput. Methods Appl. Mech. Eng. 191, 4721–4746 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eskilsson, C., Sherwin, S.J.: A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations. Int. J. Numer. Methods Fluids 45, 605–623 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ern, A., Piperno, S., Djadel, K.: A well-balanced Runge–Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying. Int. J. Numer. Methods Fluids 58, 1–25 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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, 5587–5609 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gallardo, J.M., Parés, C., Castro, M.: On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J. Comput. Phys. 227, 574–601 (2007)

    Article  MathSciNet  MATH  Google 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, A1233–A1253 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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 (2016)

    MathSciNet  MATH  Google Scholar 

  23. Giraldo, F.X., Hesthaven, J.S., Warburton, T.: Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations. J. Comput. Phys. 181, 499–525 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2007)

    MATH  Google Scholar 

  25. Hou, S., Liu, X.-D.: Solutions of multi-dimensional hyperbolic systems of conservation laws by square entropy condition satisfying discontinuous Galerkin method. J. Sci. Comput. 31, 127–151 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kopriva, D.A., Gassner, G.: On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods. J. Sci. Comput. 44, 136–155 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kurganov, A., Levy, D.: Central-upwind schemes for the Saint–Venant system. Math. Model. Numer. Anal. 36, 397–425 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. LeVeque, R.J.: Balancing source terms and flux gradients on high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146, 346–365 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Ranocha, H.: Shallow water equations: split-form, entropy stable, well-balanced, and positivity preserving numerical methods. Int. J. Geomath. 8, 85–133 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Schwanenberg, D., Köngeter, J.: A discontinuous Galerkin method for the shallow water equations with source terms. In: Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, Part I: Overview, pp. 289–309. Springer, Berlin (2000)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. 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 

  34. Tadmor, E.: Entropy stable schemes. In: Abgrall, R., Shu, C.-W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems, vol. XVII, pp. 467–493. Elsevier, London (2016)

    Google Scholar 

  35. Wen, X., Gao, Z., Don, W.S., Xing, Y., Li, P.: Application of positivity-preserving well-balanced discontinuous Galerkin method in computational hydrology. Comput. Fluids 139, 112–119 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wintermeyer, N., Winters, A.R., Gassner, G.J., Warburton, T.: An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs. J. Comput. Phys. 375, 447–480 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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. J. Comput. Phys. 340, 200–242 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Xing, Y.: Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium. J. Comput. Phys. 257, 536–553 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Xing, Y., Shu, C.-W.: High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208, 206–227 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Xing, Y., Shu, C.-W.: High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys. 214, 567–598 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  41. Xing, Y., Shu, C.-W.: A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. Commun. Comput. Phys. 1, 100–134 (2006)

    MATH  Google Scholar 

  42. Xing, Y., Shu, C.-W.: High-order finite volume WENO schemes for the shallow water equations with dry states. Adv. Water Resour. 34, 1026–1038 (2011)

    Article  Google 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, 1476–1493 (2010)

    Article  Google Scholar 

  44. Xing, Y., Zhang, X.: Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes. J. Sci. Comput. 57, 19–41 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091–3120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work of X. Wen is supported by the China Scholarship Council fellowship. The work of Y. Xing is partially sponsored by NSF Grant DMS-1753581. The work of X. Wen, Z. Gao and W.S. Don is partially supported by the National Natural Science Foundation of China (11871443), Shandong Provincial Natural Science Foundation (ZR2017MA016), and Shandong Provincial Qingchuang Science and Technology Project (2019KJI002). W.S. Don also likes to thank the Ocean University of China for providing the startup funding (201712011) that is used in supporting this work.

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Appendix A: Two-Dimensional DG Methods

Appendix A: Two-Dimensional DG Methods

The two-dimensional entropy stable DG methods (4.5), after following the definition of the derivative operators \(D_h^x\) in Definition 3 and the entropy conservative numerical fluxes (4.3) and (4.4), can be expanded as

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \sum _k \left( \frac{\partial h}{\partial t},\omega \right) _{\Omega ^k}+\left( \frac{\partial {\mathbb {P}}m_{1,e}}{\partial x}+\frac{\partial {\mathbb {P}}m_{2,e}}{\partial y},\omega \right) _{\Omega ^k} \\ \displaystyle \\ \displaystyle \qquad +\left\langle f_S^{(1)}\left( Q_e^+,Q_e\right) -{\mathbb {P}}m_{1,e},\omega n_x\right\rangle _{\partial \Omega ^k} + \left\langle g_S^{(1)}\left( Q_e^+,Q_e\right) -{\mathbb {P}}m_{2,e},\omega n_y\right\rangle _{\partial \Omega ^k} = 0, \\ \displaystyle \\ \displaystyle \sum _k \left( \frac{\partial m_1}{\partial t},\omega \right) _{\Omega ^k} +\left( \frac{1}{2}\frac{\partial }{\partial x}\left( {\mathbb {P}}(m_{1,e}u_e)\right) +\frac{1}{2}u_e\frac{\partial }{\partial x}\left( {\mathbb {P}}m_{1,e}\right) +\frac{1}{2}m_{1,e}\frac{\partial }{\partial x} u_e +gh_e\frac{\partial }{\partial x}\left( {\mathbb {P}}h_e\right) ,\omega \right) _{\Omega ^k} \\ \displaystyle \\ \displaystyle \qquad +\left( gh_e\frac{\partial }{\partial x}b_e,\omega \right) _{\Omega ^k} +\left( \frac{1}{2}\frac{\partial }{\partial y}\left( ({\mathbb {P}}m_{2,e})u_e\right) +\frac{1}{2}u_e\frac{\partial }{\partial y}\left( {\mathbb {P}}m_{2,e}\right) +\frac{1}{2}m_{2,e}\frac{\partial }{\partial y}u_e,\omega \right) _{\Omega ^k} \\ \displaystyle \\ \displaystyle \qquad +\left\langle f_S^{(2)}\left( Q_e^+,Q_e\right) +\hbox {I},\omega n_x\right\rangle _{\partial \Omega ^k} +\hbox {V} +\left\langle g_S^{(2)}\left( Q_e^+,Q_e\right) +\hbox {II},\omega n_y\right\rangle _{\partial \Omega ^k} +\hbox {VI} =0, \\ \displaystyle \\ \displaystyle \sum _k \left( \frac{\partial m_2}{\partial t},\omega \right) _{\Omega ^k} +\left( \frac{1}{2}\frac{\partial }{\partial x}\left( ({\mathbb {P}}m_{1,e})v_e\right) +\frac{1}{2}v_e\frac{\partial }{\partial x}\left( {\mathbb {P}}m_{1,e}\right) +\frac{1}{2}m_{1,e}\frac{\partial }{\partial x}v_e,\omega \right) _{\Omega ^k} \\ \displaystyle \\ \displaystyle \quad +\left( \frac{1}{2}\frac{\partial }{\partial y}\left( {\mathbb {P}}\left( m_{2,e}v_e\right) \right) +\frac{1}{2}v_e\frac{\partial }{\partial y}\left( {\mathbb {P}}m_{2,e}\right) +v\frac{1}{2}m_{2,e}\frac{\partial }{\partial y}v_e +gh_e\frac{\partial }{\partial y} \left( {\mathbb {P}}h_e\right) ,\omega \right) _{\Omega ^k} +\left( gh_e\frac{\partial }{\partial y}b_e,\omega \right) _{\Omega ^k} \\ \displaystyle \\ \displaystyle \qquad +\left\langle f_S^{(3)}\left( Q_e^+,Q_e\right) + \hbox {III},\omega n_y\right\rangle _{\partial \Omega ^k} + \hbox {VII} +\left\langle g_S^{(3)}\left( Q_e^+,Q_e\right) +\hbox {IV},\omega n_y\right\rangle _{\partial \Omega ^k} +\hbox {VIII} =0, \end{array} \right. \end{aligned}$$

with the terms \(\hbox {I}\) - \(\hbox {VIII}\) defined by

$$\begin{aligned} \hbox {I}&=\Phi (m_{1,e},u_e) -\frac{1}{2}gh_e\left( {\mathbb {P}}h_e - \llbracket b_e\rrbracket \right) ,\qquad \hbox {II}=\Phi (m_{2,e},u_e),\\ \hbox {III}&=\Phi (m_{1,e},v_e), \qquad \hbox {IV}=\Phi (m_{2,e},v_e) -\frac{1}{2}gh_e\left( {\mathbb {P}}h_e -\llbracket b_e\rrbracket \right) , \\ \hbox {V}&=\frac{1}{4}\Psi _x(m_{1,e},u_e)+\frac{1}{2}g\Psi _x(h_e,h_e),\qquad \hbox {VI}=\frac{1}{4}\Psi _x(m_{2,e},u_e),\\ \hbox {VII}&=\frac{1}{4}\Psi _y(m_{1,e},v_e),\qquad \hbox {VIII}=\frac{1}{4}\Psi _y(m_{2,e},v_e)+\frac{1}{2}g\Psi _y(h_e,h_e), \end{aligned}$$

where the notations

$$\begin{aligned} \Phi (a,b)= & {} -\frac{1}{2}{\mathbb {P}} (ab) +\frac{1}{4}({\mathbb {P}}a+a)b, \quad \Psi _x(a,b)=\langle {\mathcal {E}}(a),{\mathbb {P}}(b\omega )n_x\rangle _{\partial \Omega ^k}, \quad \Psi _y(a,b)\\= & {} \langle {\mathcal {E}}(a),{\mathbb {P}}(b\omega )n_y\rangle _{\partial \Omega ^k}, \end{aligned}$$

are used. An equivalent form of the DG methods, which uses the vector variables and local matrices to guide the efficient implementation, is available in [9].

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Wen, X., Don, W.S., Gao, Z. et al. Entropy Stable and Well-Balanced Discontinuous Galerkin Methods for the Nonlinear Shallow Water Equations. J Sci Comput 83, 66 (2020). https://doi.org/10.1007/s10915-020-01248-3

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