Skip to main content
SpringerLink
Log in

Convergence of higher order finite volume schemes on irregular grids

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We prove convergence to the entropy solution of a general class of higher order finite volume schemes on unstructured, irregular grids for multidimensional scalar conservation laws. Such grids allow for cells to become flat in the limit. We derive a new entropy inequality for higher order schemes built on Godunov’s numerical flux. Our result implies convergence of suitably modified versions of MUSCL-type finite volume schemes, ENO schemes and the discontinuous Galerkin finite element method.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Boris and D. Book, Flux corrected transport I. SHASTA, a fluid transport algorithm that works, J. Comp. Phys. 11 (1973) 38–69.

    Article  MATH  Google Scholar 

  2. B. Cockburn, Quasimonotone schemes for scalar conservation laws I, SIAM J. Numer. Anal. 26 (1989) 1325–1341.

    Article  MathSciNet  MATH  Google Scholar 

  3. B. Cockburn, F. Coquel and P. LeFloch, An error estimate for finite volume methods for scalar conservation laws, Preprint (1991).

  4. B. Cockburn, F. Coquel and P. LeFloch, An error estimate for high-order accurate finite volume methods for scalar conservation laws, Preprint (1991).

  5. B. Cockburn, F. Coquel, P. LeFloch and C. W. Shu, Convergence of finite volume methods, IMA Preprint #771 (1991).

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

    MathSciNet  MATH  Google Scholar 

  7. P. Collela and P. Woodward, The piecewise parabolic method for gas-dynamical simulations, J. Comp. Phys. 54 (1984) 174–201.

    Article  Google Scholar 

  8. F. Coquel and P. LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory, SIAM J. Numer. Anal. 30 (1993) 675–700.

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Coquel and P. LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach, Math. Comp. 57 (1991) 169–210.

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Coquel and P. LeFloch, The finite volume method on general triangulations converges for general conservation laws, Preprint (1992).

  11. M. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980) 1–21.

    Article  MathSciNet  MATH  Google Scholar 

  12. R. DiPerna, Measure valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985) 223–270.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Diff. Eq. 71 (1988) 93–122.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Geiben, Convergence of MUSCL-type upwind finite volume schemes on unstructured triangular grids, SFB256 Preprint no. 318, Bonn, Germany (1993).

  15. S. Godunov, Finite difference methods for numerical computation of discontinuous solutions of the equations of fluid dynamics, Math. Sb. 47 (1959) 271–295.

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. A. Harten, B. Engquist, S. Osher and S. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes III, J. Comp. Phys. 71 (1987) 231–303.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Jaffre, J. Johnson and A. Szepessy, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Preprint No. 1993-11/ISSN 0347-2809, Chalmers University of Technology, Sweden (1993).

    Google Scholar 

  19. D. Kröner, S. Noelle and M. Rokyta, Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in two space dimensions, SFB256 Preprint no. 268, Bonn, Germany (1993).

  20. D. Kröner and M. Rokyta, Convergence of upwind finite volume schemes for scalar conservation laws in 2-D, SFB256 Preprint no. 208, Bonn, Germany (1992), to appear in SIAM J. Numer. Anal. (1994).

  21. S. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR-Sbornik 10 (1970) 217–243.

    Article  Google Scholar 

  22. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Comp. Math. and Math. Phys. 16, no. 6 (1976) 105–119.

    Article  Google Scholar 

  23. P. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7 (1954) 159–193.

    Article  MathSciNet  MATH  Google Scholar 

  24. P. Lax, Shock waves and entropy,Proc. Symp., University of Wisconsin, ed. E.H. Zarantonello (1971) pp. 603–634.

  25. A. Leroux, Convergence of an accurate scheme for first order quasi linear equations, RAIRO Numer. Anal. 15 (1981) 151–170.

    MathSciNet  Google Scholar 

  26. X-D. Liu, A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws, SIAM J. Numer. Anal. 30 (1993) 701–716.

    Article  MathSciNet  MATH  Google Scholar 

  27. B. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986) 59–69.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Majda and S. Osher, Numerical viscosity and the entropy condition, Comm. Pure Appl. Math. 32 (1979) 797–838.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comp. Phys. 87 (1990) 408–463.

    Article  MathSciNet  MATH  Google Scholar 

  30. S. Osher, Riemann solvers, the entropy condition and difference approximations, SIAM J. Numer. Anal. 21 (1984) 217–235.

    Article  MathSciNet  MATH  Google Scholar 

  31. S. Osher and R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comp. 41 (1983) 321–336.

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws, Math. Comp. 50 (1988) 19–51.

    Article  MathSciNet  MATH  Google Scholar 

  33. R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983) 91–106.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Szepessy, Convergence of a shock capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions, Math. Comp. 53 (1989) 527–545.

    Article  MathSciNet  MATH  Google Scholar 

  35. E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp. 43 (1984) 369–381.

    Article  MathSciNet  MATH  Google Scholar 

  36. L. Tartar, The compensated compactness method applied to systems of conservation laws, in:Systems of Nonlinear PDE, ed. J. Ball, NATO ASI Series (1983) pp. 263–285.

  37. B. van Leer, Towards the ultimate conservative difference scheme V, J. Comp. Phys. 32 (1979) 101–136.

    Article  Google Scholar 

  38. J. Vila, High order schemes and entropy conditions for nonlinear hyperbolic systems of conservation laws, Math. Comp. 50 (1988) 53–73.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Vila, Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws: I. Explicit monotone schemes, Preprint (1993).

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Bonn University, Wegeler St. 10, D-53115, Bonn, Germany

    Sebastian Noelle

Authors
  1. Sebastian Noelle
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by E. Tadmor

Supported by Deutsche Forschungsgemeinschaft, SFB256.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Noelle, S. Convergence of higher order finite volume schemes on irregular grids. Adv Comput Math 3, 197–218 (1995). https://doi.org/10.1007/BF02431999

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02431999

Keywords

AMS subject classification

Search

Navigation