Skip to main content
Springer Nature Link
Log in

A High-Order Well-Balanced Positivity-Preserving Moving Mesh DG Method for the Shallow Water Equations With Non-Flat Bottom Topography

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A rezoning-type adaptive moving mesh discontinuous Galerkin method is proposed for the numerical solution of the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the simulation of perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance and positivity-preserving properties, strategies are discussed in the use of slope limiting, positivity-preservation limiting, and data transferring between meshes. Particularly, it is suggested that a DG-interpolation scheme be used for the interpolation of both the flow variables and bottom topography from the old mesh to the new one and after each application of the positivity-preservation limiting on the water depth, a high-order correction be made to the approximation of the bottom topography according to the modifications in the water depth. Mesh adaptivity is realized using a moving mesh partial differential equation and a metric tensor based on the equilibrium variable and water depth. A motivation for the latter is to adapt the mesh according to both the perturbations of the lake-at-rest steady state and the water depth distribution. Numerical examples in one and two spatial dimensions are presented to demonstrate the well-balance and positivity-preserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady state. They also show that the mesh adaptation based on the equilibrium variable and water depth give more desirable results than that based on the commonly used entropy function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
€32.70 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30

Similar content being viewed by others

References

  1. Arpaia, L., Ricchiuto, M.: r-adaptation for shallow water flows: conservation, well balancedness, efficiency. Comput. Fluids 160, 175–203 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alcrudo, F., Benkhaldoun, F.: Exact solutions to the Riemann problem of the shallow water equations with a bottom step. Comput. Fluids 30, 643–671 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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 

  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. Cheng, J., Shu, C.-W.: A high order ENO conservative Lagrangian type scheme for the compressible Euler equations. J. Comput. Phys. 227, 1567–1596 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. 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, 90–113 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection dominated problems. J. Sci. Comput. 16, 173–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chinnayya, A., LeRoux, A., Seguin, N.: A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon. Int. J. Finite 1, 1–33 (2004)

    MathSciNet  Google Scholar 

  11. Donat, R., Martí, M.C., Martínez-Gavara, A., Mulet, P.: Well-balanced adaptive mesh refinement for shallow water flows. J. Comput. Phys. 257, 937–953 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. 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. Meth. Fluids 58, 1–25 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Goodman, J.B., LeVeque, R.J.: On the accuracy of stable schemes for 2D scalar conservation laws. Math. Comp. 45, 15–21 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, W., Ren, Y., Russell, R.: Moving mesh methods based on moving mesh partial differential equations. J. Comput. Phys. 113, 279–290 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, W., Ren, Y., Russell, R.: Moving mesh partial differential equations (MMPDEs) based upon the equidistribution principle. SIAM J. Numer. Anal. 31, 709–730 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang, W.: Variational mesh adaptation: isotropy and equidistribution. J. Comput. Phys. 174, 903–924 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huang, W., Sun, W.: Variational mesh adaptation II: error estimates and monitor functions. J. Comput. Phys. 184, 619–648 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang, W.: Mathematical principles of anisotropic mesh adaptation. Comm. Comput. Phys. 1, 276–310 (2006)

    MATH  Google Scholar 

  20. Huang, W., Russell, R.: Adaptive Moving Mesh Methods. Applied Mathematical Sciences Series. Springer: New York 174, (2011)

  21. Huang, W., Kamenski, L.: A geometric discretization and a simple implementation for variational mesh generation and adaptation. J. Comput. Phys. 301, 322–337 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huang, W., Kamenski, L.: On the mesh nonsingularity of the moving mesh PDE method. Math. Comp. 87, 1887–1911 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, X.-D., Osher, S.: Non-oscillatory high order accurate self similar maximum principle satisfying shock capturing schemes. SIAM J. Numer. Anal. 33, 760–779 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lamby, P., Müller, S., Stiriba, Y.: Solution of shallow water equations using fully adaptive multiscale schemes. Int. J. Numer. Meth. Fluids 49, 417–437 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, R., Tang, T.: Moving mesh discontinuous Galerkin method for hyperbolic conservation laws. J. Sci. Comput. 27, 347–363 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, G., Song, L., Gao, J.: High order well-balanced discontinuous Galerkin methods based on hydrostatic reconstruction for shallow water equations. J. Comput. App. Math. 340, 546–560 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Luo, D., Huang, W., Qiu, J.: A quasi-Lagrangian moving mesh discontinuous Galerkin method for hyperbolic conservation laws. J. Comput. Phys. 396, 544–578 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, G., Lu, C., Qiu, J.: Hybrid well-balanced WENO schemes with different indicators for shallow water equations. J. Sci. Comput. 51, 527–559 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mavriplis, D., Yang, Z.: Construction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes. J. Comput. Phys. 213, 557–573 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mavriplis, D., Nastase, C.: On the geometric conservation law for high-order discontinuous Galerkin discretizations on dynamically deforming meshes. J. Comput. Phys. 230, 4285–4300 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Noelle, S., Xing, Y., Shu, C.-W.: High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226, 29–58 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Remacle, J.-F., Frazao, S.S., Li, X., Shephard, M.: Adaptive discontinuous Galerkin method for the shallow water equations. Int. J. Numer. Meth. Fluids 52, 903–923 (2006)

    Article  MATH  Google Scholar 

  34. Ricchiuto, M., Bollermann, A.: Stabilized residual distribution for shallow water simulations. J. Comput. Phys. 228, 1071–1115 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tang, H., Tang, T.: Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws. SIAM J. Numer. Anal. 41, 487–515 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tang, H.: Solution of the shallow-water equations using an adaptive moving mesh method. Int. J. Numer. Meth. Fluids 44, 789–810 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tumolo, G., Bonaventura, L., Restelli, M.: A semi-implicit, semi-lagrangian, p-adaptive discontinuous Galerkin method for the shallow water equations. J. Comput. Phys. 232, 46–67 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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 

  39. 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 

  40. 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. Comm. Comput. Phys. 1, 100–134 (2006)

    MATH  Google Scholar 

  41. Xing, Y., Zhang, X., Shu, C.-W.: Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resourc. 33, 1476–1493 (2010)

    Article  Google Scholar 

  42. 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 

  43. 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 

  44. Zhang, M., Cheng, J., Huang, W., Qiu, J.: An adaptive moving mesh discontinuous Galerkin method for the radiative transfer equation. Comm. Comput. Phys. 27, 1140–1173 (2020)

    Article  MathSciNet  Google Scholar 

  45. Zhang, M., Huang, W., Qiu, J.: High-order conservative positivity-preserving DG-interpolation for deforming meshes and application to moving mesh DG simulation of radiative transfer. SIAM J. Sci. Comput. 42, A3109–A3135 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhang, Z., Naga, A.: A new finite element gradient recovery method: Superconvergence property. SIAM J. Sci. Comput. 26, 1192–1213 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhou, F., Chen, G., Noelle, S., Guo, H.: A well-balanced stable generalized Riemann problem scheme for shallow water equations using adaptive moving unstructured triangular meshes. Int. J. Numer. Meth. Fluids 73, 266–283 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Zhou, J.G., Causon, D.M., Mingham, C.G., Ingram, D.M.: The surface gradient method for the treatment of source terms in the shallow-water equations. J. Comput. Phys. 168, 1–25 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zhang, X., Shu, C.-W.: On positivity preserving high order discontinuous Galerkin methods for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918–8934 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  50. Zhang, X., Xia, Y., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving High Order discontinuous Galerkin schemes for conservation Laws on Triangular Meshes. J. Sci. Comput. 50, 29–62 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, China

    Min Zhang

  2. Department of Mathematics, University of Kansas, Lawrence, Kansas, 66045, USA

    Weizhang Huang

  3. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, Fujian, 361005, China

    Jianxian Qiu

Authors
  1. Min Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weizhang Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianxian Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianxian Qiu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

M. Zhang and J. Qiu were supported partly by National Natural Science Foundation (China) grant 12071392 and Science Challenge Project (China), No. TZ 2016002 . This work was carried out while M. Zhang was visiting the Department of Mathematics, the University of Kansas under the support by the China Scholarship Council (CSC: 201806310065).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, M., Huang, W. & Qiu, J. A High-Order Well-Balanced Positivity-Preserving Moving Mesh DG Method for the Shallow Water Equations With Non-Flat Bottom Topography. J Sci Comput 87, 88 (2021). https://doi.org/10.1007/s10915-021-01490-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01490-3

Keywords

Mathematics Subject Classification

Search

Navigation