Journal of Scientific Computing

, Volume 48, Issue 1–3, pp 141–163 | Cite as

Two-Dimensional Compact Third-Order Polynomial Reconstructions. Solving Nonconservative Hyperbolic Systems Using GPUs

  • José M. Gallardo
  • Sergio Ortega
  • Marc de la Asunción
  • José Miguel Mantas


We present a new kind of high-order reconstruction operator of polynomial type, which is used in combination with the scheme presented in Castro et al. (J. Sci. Comput. 39:67–114, 2009) for solving nonconservative hyperbolic systems. The implementation of the scheme is carried out on Graphics Processing Units (GPUs), thus achieving a substantial improvement of the speedup with respect to normal CPUs. As an application, the two-dimensional shallow water equations with geometrical source term due to the bottom slope is considered.


Finite volume methods Shallow water Nonconservative hyperbolic systems High-order schemes Well-balanced GPUs 


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  1. 1.
    Abgrall, R.: An essentially non-oscillatory reconstruction procedure on finite-element type meshes: Application to compressible flows. Comput. Methods Appl. Mech. Eng. 116, 95–101 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abgrall, R.: On essentially non-oscillatory schemes on unstructured meshes: Analysis and implementation. J. Comput. Phys. 114, 45–58 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Castro, M.J., Gallardo, J.M., Parés, C.: High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math. Comput. 75, 1103–1134 (2006) zbMATHCrossRefGoogle Scholar
  4. 4.
    Castro, M.J., Fernández, E.D., Ferreiro, A.M., García, A., Parés, C.: High order extension of Roe schemes for two dimensional nonconservative hyperbolic systems. J. Sci. Comput. 39, 67–114 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dumbser, M., Käser, M.: Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys. 221, 693–723 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Friedrich, O.: Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comput. Phys. 144, 194–212 (1998) CrossRefMathSciNetGoogle Scholar
  7. 7.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hagen, T.R., Hjelmervik, J.M., Lie, K.A., Natvig, J.R., Ofstad, M.: Visual simulation of shallow-water waves. Simul. Model. Pract. Theory 13, 716–726 (2005) CrossRefGoogle Scholar
  9. 9.
    Harten, A., Hyman, J.M.: Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 50, 235–269 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hu, C., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jiang, G., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lastra, M., Mantas, J.M., Ureña, C., Castro, M.J., García, J.A.: Simulation of shallow-water systems using graphics processing units. Math. Comput. Simul. 80, 598–618 (2009) zbMATHCrossRefGoogle Scholar
  13. 13.
    Liu, X.D., Osher, S., Chan, T.: Weighted essentially nonoscillatory schemes. J. Comput. Phys. 115, 200–212 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    de la Asunción, M., Mantas, J.M., Castro, M.J.: Simulation of one-layer shallow water systems on multicore and CUDA architectures. J. Supercomput. (2009). doi: 10.1007/s11227-010-0406-2 Google Scholar
  15. 15.
    Marquina, A.: Local piecewise hyperbolic reconstructions for nonlinear scalar conservation laws. SIAM J. Sci. Comput. 15, 892–915 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Noelle, S., Pankratz, N., Puppo, G., Natvig, J.: Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213, 474–499 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
  18. 18.
    NVIDIA. CUDA Zone. Accessed November 2009
  19. 19.
    Parés, C.: Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44, 300–321 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Owens, J.D., Luebke, D., Govindaraju, N., Harris, M., Krüger, J., Lefohn, A.E., Purcell, T.: A Survey of General-Purpose Computation on Graphics Hardware, Eurographics 2005 State of the Art Report (2005) Google Scholar
  21. 21.
    Rumpf, M., Strzodka, R.: Graphics processor units: new prospects for parallel computing. Lect. Notes Comput. Sci. Eng. 51, 89–121 (2006) CrossRefGoogle Scholar
  22. 22.
    Schroll, H.J., Svensson, F.: A bi-hyperbolic finite volume method on quadrilateral meshes. J. Sci. Comput. 26, 237–260 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. ICASE Report n. 97–65 (1997) Google Scholar
  24. 24.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–71 (1998) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Walz, G.: Romberg type cubature over arbitrary triangles. Mannheimer Mathem. Manuskripte Nr. 225, Mannhein (1997) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • José M. Gallardo
    • 1
  • Sergio Ortega
    • 1
  • Marc de la Asunción
    • 1
  • José Miguel Mantas
    • 2
  1. 1.Department of Mathematical AnalysisUniversity of MálagaMálagaSpain
  2. 2.Software Engineering DepartmentUniversity of GranadaGranadaSpain

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