An Entropy-Stable Residual Distribution Scheme for the System of Two-Dimensional Inviscid Shallow Water Equations

  • Wei Shyang Chang
  • Farzad IsmailEmail author
  • Hossain Chizari


An entropy-stable residual distribution (RD) method is developed for the systems of two-dimensional shallow water equations (SWE). The construction of entropy stability for the residual distribution method is derived from finite volume method principles, albeit using a multidimensional approach. The paper delves into in-depth discussions on how finite volume methods achieve entropy stability in a “one-dimensional” sense for two-dimensional systems of SWE unlike the residual distribution methods. Results herein demonstrate the superiority of the entropy-stable RD methods relative to their finite volume counterparts, especially on highly irregular triangular grids. The comparative results with other established RD methods are also included, depicting similar performances with the Lax–Wendroff method for unsteady smooth flows but more accurate than the multidimensional upwind approaches (N, LDA) on both smooth and discontinuous test cases.


Entropy-stable Residual distribution Semi-discrete Shallow water Flux function 

Mathematics Subject Classification




We would like to thank Universiti Sains Malaysia for financially supporting this research work under the Academic Staff Training Scheme (ASTS) and the USM Research University Grant (No: 1001/PAERO/8014091).

Supplementary material


  1. 1.
    Abgrall, R.: Residual distribution schemes: current status and future trends. Comput. Fluids 35(7), 641–669 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abgrall, R.: A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes. J. Comput. Phys. 372, 640–666 (2018)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Abgrall, R., Mezine, M.: Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow. J. Comput. Phys. 188(1), 16–55 (2003)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Abgrall, R., Roe, P.: High-order fluctuation schemes on triangular meshes. J. Sci. Comput. 19(1), 3–36 (2003)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Barth, T.: An introduction to recent developments in theory and numerics of conservation laws. In: Numerical Methods for Gas-Dynamic Systems on Unstructured Meshes (1999)Google Scholar
  6. 6.
    Chandrashekar, P.: Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier–Stokes equations. Commun. Comput. Phys. 14(5), 1252–1286 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chizari, H., Ismail, F.: Accuracy variations in residual distribution and finite volume methods on triangular grids. Bull. Malays. Math. Sci. Soc. 22, 1–34 (2015)zbMATHGoogle Scholar
  8. 8.
    Chizari, H., Ismail, F.: A grid-insensitive LDA method on triangular grids solving the system of Euler equations. J. Sci. Comput. 71(2), 839–874 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chizari, H., Singh, V., Ismail, F.: Developments of entropy-stable residual distribution methods for conservation laws II: system of Euler equations. J. Comput. Phys. 330, 1093–1115 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Csík, A., Ricchiuto, M., Deconinck, H.: A conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws. J. Comput. Phys. 179(1), 286–312 (2002)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Deconinck, H., Sermeus, K., Abgrall, R.: Status of multidimensional upwind residual distribution schemes and applications in aeronautics. In: Fluids Conference, 20002328. AIAA Conference (2000)Google Scholar
  12. 12.
    Dobeš, J., Deconinck, H.: Second order blended multidimensional upwind residual distribution scheme for steady and unsteady computations. J. Comput. Appl. Math. 215(2), 378–389 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dubey, R., Biswas, B.: Suitable diffusion for constructing non-oscillatory entropy stable schemes. J. Comput. Phys. 372, 912–930 (2018)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fischer, T., Carpenter, M.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252(1), 518–557 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Fjordholm, U., 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)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Fjordholm, U., Mishra, S., Tadmor, E.: Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50(2), 544–573 (2012)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Garcia-Navarro, P., Hubbard, M.E., Priestly, A.: Genuinely multidimensional for the 2d shallow water equations. J. Comput. Phys. 121(1), 79–93 (1995)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Gassner, G., Winters, A., Kopriva, D.: A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations. Appl. Math. Comput. 272, 291–308 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Guzik, S., Groth, C.: Comparison of solution accuracy of multidimensional residual distribution and Godunov-type finite-volume methods. Int. J. Comput. Fluid Dyn. 22, 61–83 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hubbard, M., Baines, M.: Conservative multidimensional upwinding for the steady two dimensional shallow water equations. J. Comput. Phys. 138, 419–448 (1997)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hughes, T., Franca, L., Mallet, M.: A new finite element formulation for compressible fluid dynamics: I. Symmetric forms of the compressible Euler and Navier Stokes equations and the second law of thermodynamics. Comput. Methods Appl. Mech. Eng. 54, 223–234 (1986)zbMATHGoogle Scholar
  22. 22.
    Ismail, F.: Toward a reliable prediction of shocks in hypersonic flow: resolving carbuncles with entropy and vorticity control. Ph.D. thesis, The University of Michigan (2006)Google Scholar
  23. 23.
    Ismail, F., Chang, W.S., Chizari, H.: On flux-difference residual distribution methods. Bull. Malays. Math. Sci. Soc. 41(3), 1629–1655 (2017)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ismail, F., Chizari, H.: Developments of entropy-stable residual distribution methods for conservation laws I: Scalar problems. J. Comput. Phys. 330, 1093–1115 (2017)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions II: entropy production at shocks. J. Comput. Phys. 228(15), 5410–5436 (2009)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Merriam, M.: An entropy based approach to nonlinear stability. NASA TM-101086, Ames Research Center (1989)Google Scholar
  27. 27.
    Ricchiuto, M.: Contributions to the development of residual discretizations for hyperbolic conservation laws with application to shallow water flows. HDR thesis (2011)Google Scholar
  28. 28.
    Ricchiuto, M., Abgrall, R., Deconinck, H.: Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes. J. Comput. Phys. 222(1), 287–331 (2007)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Ricchiuto, M., Bollermann, A.: Stabilized residual distribution for shallow water simulations. J. Comput. Phys. 228(4), 1071–1115 (2009)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Sermeus, K., Deconinck, H.: An entropy fix for multi-dimensional upwind residual distribution schemes. Comput. Fluids 34, 617–640 (2005)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Singh, V., Chizari, H., Ismail, F.: Non-unified compact residual-distribution methods for scalar advection–diffusion problems. J. Sci. Comput. 76(3), 1521–1546 (2018)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Tadmor, E.: Skew-self adjoint form for systems of conservation laws. J. Math. Anal. Appl. 103(2), 428–442 (1984)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Tadmor, E.: Entropy functions for symmetric systems of conservation laws. J. Math. Anal. Appl. 122(2), 355–359 (1987)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. Math. Comput. 49(179), 91–103 (1987)MathSciNetzbMATHGoogle Scholar
  35. 35.
    van der Weide, E.: Compressible flow simulation on unstructured grids using multidimensional upwind schemes. Ph.D. thesis, Delft University of Technology (1998)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  • Wei Shyang Chang
    • 1
  • Farzad Ismail
    • 1
    Email author
  • Hossain Chizari
    • 2
  1. 1.School of Aerospace Engineering, Engineering CampusUniversiti Sains MalaysiaPulau PinangMalaysia
  2. 2.International Center for Applied Mechanics (ICAM), State Key Laboratory for Strength and Vibration of Mechanical StructuresXi’an Jiaotong University (XJTU)Xi’anPeople’s Republic of China

Personalised recommendations