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Well-Balanced Discontinuous Galerkin Method for Shallow Water Equations with Constant Subtraction Techniques on Unstructured Meshes

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Abstract

The classical Saint–Venant shallow water equations on complex geometries have wide applications in many areas including coastal engineering and atmospheric modeling. The main numerical challenge in simulating Saint–Venant equations is to maintain the high order of accuracy and well-balanced property simultaneously. In this paper, we propose a high-order accurate and well-balanced discontinuous Galerkin (DG) method on two dimensional (2D) unstructured meshes for the Saint–Venant shallow water equations. The technique used to maintain well-balanced property is called constant subtraction and proposed in Yang et al. (J Sci Comput 63:678–698, 2015). Hierarchical reconstruction limiter with a remainder correction technique is introduced to control numerical oscillations. Numerical examples with smooth and discontinuous solutions are provided to demonstrate the performance of our proposed DG methods.

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References

  1. Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. Wiley, New York (1989)

    MATH  Google Scholar 

  2. Barré de Saint-Venant, A.J.C.: Théorie du mouvement non permanent des eaux, avec application aux crues de rivières et à l’introduction des marées dans leur lit. C.R. Acad. Sci. Paris, 73:237–240 (1871)

  3. Behrens, J.: Atmospheric and ocean modeling with an adaptive finite element solver for the shallow-water equations. Appl. Numer. Math. 26(1–2), 217–226 (1998)

    MathSciNet  MATH  Google Scholar 

  4. Biswas, R., Devine, K.D., Flaherty, J.E.: Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math. 14(1–3), 255–283 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Bryson, S., Epshteyn, Y., Kurganov, A., Petrova, G.: Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system. ESAIM Math. Modell. Numer. Anal. 45(3), 423–446 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Burbeau, A., Sagaut, P., Bruneau, C.-H.: A problem-independent limiter for high-order Runge–Kutta discontinuous Galerkin methods. J. Comput. Phys. 169(1), 111–150 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Caleffi, V., Valiani, A.: Well-balanced bottom discontinuities treatment for high-order shallow water equations WENO scheme. J. Eng. Mech. 135(7), 684–696 (2009)

    Google Scholar 

  8. Caleffi, V., Valiani, A., Bernini, A.: Fourth-order balanced source term treatment in central WENO schemes for shallow water equations. J. Comput. Phys. 218(1), 228–245 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Canestrelli, A., Siviglia, A., Dumbser, M., Toro, E.F.: Well-balanced high-order centred schemes for non-conservative hyperbolic systems applications to shallow water equations with fixed and mobile bed. Adv. Water Resour. 32(6), 834–844 (2009)

    Google Scholar 

  10. Chertock, A., Cui, S., Kurganov, A., Wu, T.: Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms. Int. J. Numer. Meth. Fluids 78(6), 355–383 (2015)

    MathSciNet  Google Scholar 

  11. Cockburn, B., Hou, S., Shu, C.-W.: The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54(190), 545–581 (1990)

    MathSciNet  MATH  Google Scholar 

  12. Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411–435 (1989)

    MathSciNet  MATH  Google Scholar 

  14. Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998)

    MathSciNet  MATH  Google Scholar 

  15. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes. III. In: Upwind and high-resolution schemes, pp. 218–290 (1987)

    MATH  Google Scholar 

  16. Kurganov, A., Levy, D.: Central-upwind schemes for the Saint–Venant system. ESAIM Math. Modell. Numer. Anal. 36(3), 397–425 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Kurganov, A., Petrova, G.: A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5(1), 133–160 (2007)

    MathSciNet  MATH  Google Scholar 

  18. LeBlond, P.H.: On tidal propagation in shallow rivers. J. Geophys. Res. Oceans 83(9), 4717–4721 (1978)

    Google Scholar 

  19. LeVeque, R.J.: Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146(1), 346–365 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Li, G., Xing, Y.: Well-balanced finite difference weighted essentially non-oscillatory schemes for the Euler equations with static gravitational fields. Comput. Math. Appl. 75(6), 2071–2085 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Liu, X., Albright, J., Epshteyn, Y., Kurganov, A.: Well-balanced positivity preserving central-upwind scheme with a novel wet/dry reconstruction on triangular grids for the Saint–Venant system. J. Comput. Phys. 374, 213–236 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Noelle, S., Xing, Y., Shu, C.-W.: High-order well-balanced schemes. Numerical methods for balance laws. Quaderni di Matematica 24, 1–66 (2010)

    Google Scholar 

  23. Pelinovsky, E., Kharif, C., et al.: Extreme Ocean Waves. Springer, Berlin (2008)

    MATH  Google Scholar 

  24. Qiu, J., Shu, C.-W.: Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput. 26(3), 907–929 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Rhebergen, S., Bokhove, O., van der Vegt, J.J.W.: Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys. 227(3), 1887–1922 (2008)

    MathSciNet  MATH  Google Scholar 

  26. San, O., Kara, K.: High-order accurate spectral difference method for shallow water equations. Int. J. Res. Rev. Appl. Sci. 6, 41–54 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Shu, C.-W.: TVB uniformly high-order schemes for conservation laws. Math. Comput. 49(179), 105–121 (1987)

    MathSciNet  MATH  Google Scholar 

  28. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)

    MathSciNet  MATH  Google Scholar 

  29. Walko, R.L., Avissar, R.: The ocean-land-atmosphere model (OLAM) part I: shallow-water tests. Mon. Weather Rev. 136(11), 4033–4044 (2008)

    Google Scholar 

  30. Wang, Z., Li, G., Delestre, O.: Well-balanced finite difference weighted essentially non-oscillatory schemes for the blood flow model. Int. J. Numer. Meth. Fluids 82(9), 607–622 (2016)

    MathSciNet  Google Scholar 

  31. Xing, Y.: Numerical methods for the nonlinear shallow water equations. Handb. Numer. Anal. 18, 361–384 (2017)

    MathSciNet  MATH  Google Scholar 

  32. 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(1), 206–227 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Xing, Y., Shu, C.-W.: High-order well-balanced finite difference WENO schemes for a class of hyperbolic systems with source terms. J. Sci. Comput. 27(1–3), 477–494 (2006)

    MathSciNet  MATH  Google Scholar 

  34. 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(2), 567–598 (2006)

    MathSciNet  MATH  Google Scholar 

  35. 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(1), 100–134 (2006)

    MathSciNet  MATH  Google Scholar 

  36. Xing, Y., Shu, C.-W.: A survey of high order schemes for the shallow water equations. J. Math. Study 47(3), 221–249 (2014)

    MathSciNet  MATH  Google Scholar 

  37. Xing, Y., Zhang, X.: Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes. J. Sci. Comput. 57(1), 19–41 (2013)

    MathSciNet  MATH  Google Scholar 

  38. Xu, Z., Liu, Y., Du, H., Lin, G., Shu, C.-W.: Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws. J. Comput. Phys. 230(17), 6843–6865 (2011)

    MathSciNet  MATH  Google Scholar 

  39. Xu, Z., Liu, Y., Shu, C.-W.: Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells. J. Comput. Phys. 228, 2194–2212 (2009)

    MathSciNet  MATH  Google Scholar 

  40. Yang, S., Kurganov, A., Liu, Y.: Well-balanced central schemes on overlapping cells with constant subtraction techniques for the Saint–Venant shallow water system. J. Sci. Comput. 63(3), 678–698 (2015)

    MathSciNet  MATH  Google Scholar 

  41. Zhong, X., Shu, C.-W.: A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods. J. Comput. Phys. 232(1), 397–415 (2013)

    MathSciNet  Google Scholar 

  42. Zhu, J., Qiu, J., Shu, C.-W., Dumbser, M.: Runge–Kutta discontinuous Galerkin method using WENO limiters II: unstructured meshes. J. Comput. Phys. 227(9), 4330–4353 (2008)

    MathSciNet  MATH  Google Scholar 

  43. Zhu, J., Zhong, X., Shu, C.-W., Qiu, J.: Runge–Kutta discontinuous galerkin method using a new type of WENO limiters on unstructured meshes. J. Comput. Phys. 248, 200–220 (2013)

    MathSciNet  MATH  Google Scholar 

  44. Zhu, J., Zhong, X., Shu, C.-W., Qiu, J.: Runge–Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter. Commun. Comput. Phys. 19(4), 944–969 (2016)

    MathSciNet  MATH  Google Scholar 

  45. Zhu, J., Zhong, X., Shu, C.-W., Qiu, J.: Runge–Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter on unstructured meshes. Commun. Comput. Phys. 21(3), 623–649 (2017)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588, USA

    Huijing Du

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, USA

    Yingjie Liu

  3. Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, KS, 67260, USA

    Yuan Liu

  4. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Zhiliang Xu

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  1. Huijing Du
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Correspondence to Yuan Liu.

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Huijing Du: Research supported in part by NSF Grant DMS-1853636. Yingjie Liu: Research supported in part by NSF Grants DMS-1522585 and DMS-CDS & E-MSS-1622453. Yuan Liu: Research supported in part by a grant from the Simons Foundation (426993). Zhiliang Xu: Research supported in part by NSF Grants DMS-1517293, CDS& E-MSS-1821242 and CDS & E-MSS 1854779.

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Du, H., Liu, Y., Liu, Y. et al. Well-Balanced Discontinuous Galerkin Method for Shallow Water Equations with Constant Subtraction Techniques on Unstructured Meshes. J Sci Comput 81, 2115–2131 (2019). https://doi.org/10.1007/s10915-019-01073-3

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  • DOI: https://doi.org/10.1007/s10915-019-01073-3

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