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A new theoretically motivated higher order upwind scheme on unstructured grids of simplices

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Abstract

Many approaches exist to define a cell-centered upwind finite volume scheme of higher order on an unstructured grid of simplices. However, real theoretical motivation in the form of a convergence result does not exist for these approaches. Furthermore, some theoretical results of convergence exist for higher order finite volume methods, where no description of the numerical implementation is given to realize the necessary requirements for the convergence theory. Therefore we present in this paper a new limiter function which is motivated by these requirements and ensures a convergent scheme in the theoretical context: The approximated solution converges to the entropy solution in the case of scalar conservation laws in two space dimensions. This new limiter function is combined with a typical class of reconstruction functions very efficiently, which is illustrated by several test examples for scalar conservation laws as well as systems of such laws. In connection with the requirements to be fulfilled, a proof of a maximum principle of the finite volume scheme applied to simplices and dual cells is given. So for the approach of the higher order upwind finite volume scheme on dual cells, as used in several papers, a missing proof is now given. The ideas in this proof are also applied to the discontinuous Galerkin method, so that an existing maximum principle can be improved considerably. The main advantage comes from the fact that no requirements on the discretization of the domain are necessary: no B-triangulations or Delaunay triangulation are needed.

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References

  1. M. Aftosmis, D. Gaitonde and T.S. Tavares, On the accuracy, stability and monotonicity of various reconstruction algorithms for unstructured meshes, AIAA 94-0415 (1994).

  2. E. Bänsch, An adaptive finite element strategy for the three-dimensional time-dependent Navier-Stokes equations, J. Comput. Appl. Math. 36 (1991) 3–28.

    Article  MATH  MathSciNet  Google Scholar 

  3. T.J. Barth and D.C. Jesperson, The design and application of upwind schemes on unstructured meshes, AIAA 89-0366 (1989).

  4. T.J. Barth, Computational Fluid Dynamics on Unstructured Grids and Solvers, Von Karman Institute for Fluid Dynamics Lecture Series 1990-03 (1990).

  5. M. Berzins and J.M. Ware, Positive cell-centered finite volume discretization methods for hyperbolic equations on irregular meshes, Appl. Numer. Math. (1995).

  6. A.J. Chorin, Random choice solution of hyperbolic systems, J. Comput. Phys. 22 (1976) 517–533.

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Cockburn, F. Coquel and P. LeFloch, An error estimate for high-order accurate finite volume methods for scalar conservation laws, Math. Comp. 63 (1994) 77–103.

    Article  MATH  MathSciNet  Google Scholar 

  8. 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.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. B. Cockburn and C.W. Shu, The P 1-RKDG method for two-dimensional Euler equations of gas dynamics, ICASE Report No. 91-32 (1991).

  11. E. Dick, Multigrid methods for steady Euler and Navier Stokes equations based on polynomial flux-difference splitting, in: International Series of Numerical Mathematics 98 (Birkhäuser, Basel, 1991).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. L.J. Durlofsky, B. Engquist and S. Osher, Triangle based adaptive stencils for the solution of hyperbolic conservation laws, J. Comput. Phys. 98 (1992) 64–73.

    Article  MATH  Google Scholar 

  14. B. Engquist and S. Osher, One sided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981) 321–351.

    Article  MATH  MathSciNet  Google Scholar 

  15. L. Fézoui and B. Stoufflet, A class of implicit upwind schemes for Euler simulations with unstructured meshes, J. Comput. Phys. 84 (1989) 174–206.

    Article  MATH  MathSciNet  Google Scholar 

  16. H. Fukuda and K. Yamamoto, An upwind method using unstructured triangular meshes, in: Proceedings 4th International Symposium on Computational Fluid Dynamics, Vol. 1 (1991).

  17. N.T. Frink, P. Parikh and S. Pirzadeh, A fast upwind solver for the Euler equations on three-dimensional unstructured meshes, AIAA 91-0102 (1991).

  18. M. Geiben (M. Wierse now), Numerical simulation of three-dimensional non-stationary compressible flow in complex geometries, in: Proceedings of 10th GAMM Seminar, eds. W. Hackbusch and G. Wittum, Notes on Numerical Fluid Mechanics (Vieweg, Braunschweig, 1994).

    Google Scholar 

  19. M. Geiben, D. Kröner and M. Rokyta, A Lax-Wendroff type theorem for cell-centered finite volume schemes in 2D, Preprint 278, SFB256, Bonn University (1993).

  20. N. Glinsky, L. Fézoui, M.C. Ciccoli and J.-A. Désidéri, Non-equilibrium hypersonic flow computations by implicit second-order upwind finite elements, in: Proceedings 8ht GAMM Conference on Numerical Methods in Fluid Mechanics, Notes on Numerical Fluid Mechanics 29 (Vieweg, Braunschweig, 1990).

    Google Scholar 

  21. C. Helf and U. Küster, A finite volume method with arbitrary control volumes and high order reconstruction for the Euler equations, in: Computational Fluid Dynamics '94 (Wiley, New York, 1994).

    Google Scholar 

  22. A. Harten and S.R. Chakravarthy, Multi-dimensional ENO schemes for general geometries, ICASE Report No. 91-76, NASA Contractor Report 187637.

  23. T. Ikeda, Maximum Principle on Finite Element Models for Convection-Diffusion Phenomena, Mathematics Studies 76 (North-Holland, Amsterdam, 1983).

    Google Scholar 

  24. J. Jaffre, C. Johnson and A. Szepessy, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Math. Models Methods Appl. Sci. 5(3) (1995) 367–386.

    Article  MATH  MathSciNet  Google Scholar 

  25. C. Johnson and A. Szepessy, Convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp. 49 (1987) 427–444.

    Article  MATH  MathSciNet  Google Scholar 

  26. C. Johnson, A. Szepessy and P. Hansbo, On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp. 54 (1990) 107–129.

    Article  MATH  MathSciNet  Google Scholar 

  27. D. Kröner and S. Noelle and M. Rokyta, Convergence of higher order finite volume schemes on unstructured grids for scalar conservation laws in two space dimensions, Numer. Math. 71 (1995).

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

    MATH  MathSciNet  Google Scholar 

  29. B. van Leer, Flux vector splitting for the Euler equations, in: Proceedings 8th International Conference on Numerical Methods in Fluid Dynamics (Springer, 1982).

  30. B. van Leer, Towards the ultimate conservative difference scheme IV: A new approach to numerical convection, J. Comput. Phys. 23 (1977) 276–299.

    Article  MATH  Google Scholar 

  31. B. van Leer, Towards the ultimate conservative difference scheme V: A second order sequel to Godunov's method, J. Comput. Phys. 32 (1979) 101–136.

    Article  Google Scholar 

  32. A.Y. Le Roux, Convergence of an accurate scheme for first order quasi-linear equations, RAIRO Analyse Numérique/Numerical Analysis 15 (1981) 151–170.

    MATH  MathSciNet  Google Scholar 

  33. R. LeVeque, Numerical Methods for Conservation Laws, Lecture Notes Mathematics (Birkhäuser, Basel, 1990).

    Google Scholar 

  34. S.Y. Lin, T.M. Wu and Y.S. Chin, Upwind finite volume method with a triangular mesh for conservation laws, J. Comput. Phys. 107 (1993) 324–337.

    Article  MATH  MathSciNet  Google Scholar 

  35. Xu-Dong 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  MATH  MathSciNet  Google Scholar 

  36. R. Löhner, The efficient simulation of strongly unsteady flows by the finite element method, AIAA 87-0555 (1987).

  37. D. Mavriplis, Accurate multigrid solution of Euler equations on unstructured and adaptive meshes, AIAA J. 28(2) (1990).

  38. K. Morgan, J. Peraire and J. Peiró, Unstructured grid methods for compressible flows, in: Special Course on Unstructured Grid Methods for Advection Dominated Flows, AGARD Report 787 (1992).

  39. S. Noelle, Convergence of higher order finite volume schemes on irregular grids, Adv. Comput. Math. 3(3) (1995) 197–218.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  41. J.L. Steger and R.F. Warming, Flux vector splitting of the inviscid gas dynamic equations with application to finite difference methods, J. Comput. Phys. 40 (1981) 263–293.

    Article  MATH  MathSciNet  Google Scholar 

  42. 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  MATH  MathSciNet  Google Scholar 

  43. H. Nessyahu, T. Tassa and E. Tadmor, The convergence rate of Godunov type schemes, SIAM J. Numer. Anal. 31(1) (1994).

  44. L. Tartar, Compensated compactness and applications to partial differential equations, in: Nonlinear Analysis and Mechanics, Herriot-Watts Symposium, ed. R.J. Knops, Research Notes in Mathematics 4 (Pitman Press, London, 1979).

    Google Scholar 

  45. P. Vankeirsbilck and H. Deconinck, Higher order upwind finite volume schemes with ENO properties for general unstructured meshes, in: Special Course on Unstructured Grid Methods for Advection Dominated Flows, AGARD Report 787 (1992).

  46. V. Venkatakrishnan and T.J. Barth, Application of direct solvers to unstructured meshes for the Euler and Navier-Stokes equations using upwind schemes, AIAA paper 89-0364; also: Proceedings 27th Aerospace Sciences Conference, Reno, NV (January 1992).

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

  48. D.L. Whitaker and B. Grossman, Two-dimensional Euler computations on a triangular mesh using an upwind finite volume scheme, AIAA 89-0470 (1989).

  49. M. Wierse, Higher order upwind schemes on unstructured grids for the compressible Euler equations in time-dependent geometries in 3D, Dissertation, Preprint 393, SFB256, Bonn University (1995).

  50. M. Wierse, Solving the compressible Euler equations in time-dependent geometries, in: Proceedings of the Fifth International Conference on Hyperbolic Problems: Theory, Numerics, Applications, eds. J. Glimm, M. Graham, J.W. Grove and P.J. Plohr, Stony Brook (World Scientific, 1994).

  51. M. Wierse, An automotive application on unstructured adaptive grids, in: Proceedings of ICFD Conference on Numerical Methods for Fluid Dynamics, eds. K.W. Morton and M.J. Baines, Numerical Methods for Fluid Dynamics 5 (Science Publications, Oxford, 1995).

    Google Scholar 

  52. M. Wierse and D. Kröner, Higher order upwind schemes on unstructured grids for nonstationary compressible Navier Stokes equations in complex timedependent geometries in 3D, in: Flow Simulations with High-Performance Computers II, ed. E.H. Hirschel, Notes on Numerical Fluid Mechanics 52.

  53. P.R. Woodward and P. Colella, The numerical simulation of two-dimensional flow with strong shocks, J. Comput. Phys. 54 (1984) 115–173.

    Article  MATH  MathSciNet  Google Scholar 

  54. H. Wu and S. Yang, MmB a new class of accurate high resolution scheme for conservation laws in two dimensions, Impact Comput. Sci. Engrg. 1 (1989) 217–259.

    Article  MATH  Google Scholar 

  55. S. Yang, MmB schemes on regular triangular meshes for 2D conservation laws, Preprint SC 92-17 (ZIB) (1992).

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  1. Institute for Applied Mathematics, Freiburg University, Hermann Herder Str. 10, D-79104, Freiburg, Germany

    Monika Wierse

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Wierse, M. A new theoretically motivated higher order upwind scheme on unstructured grids of simplices. Advances in Computational Mathematics 7, 303–335 (1997). https://doi.org/10.1023/A:1018955121314

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