Multiresolution schemes for conservation laws

Conference paper

Abstract

The concept of fully adaptive multiresolution finite volume schemes has been developed and investigated during the past decade. By now it has been successfully employed in numerous applications arising in engineering. In the present work a review on the methodology is given that aims to summarize the underlying concepts and to give an outlook on future developments.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Andreae, J. Ballmann, and S. Müller. Wave processes at interfaces. In G. Warnecke, editor, Analysis and numerics for conservation laws, pages 1–25. Springer, Berlin, 2005. CrossRefGoogle Scholar
  2. 2.
    A. Baeza and P. Mulet. Adaptive mesh refinement techniques for high-order shock capturing schemes for multi-dimensional hydrodynamic simulations. Int. Journal for Numerical Methods in Fluids, 52(4):455–471, 2006. MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Ballmann. Flow Modulation and Fluid-Structure-Interaction at Airplane Wings, volume 84 of Numerical Notes on Fluid Mechanics. Springer Verlag, 2003. Google Scholar
  4. 4.
    J. Ballmann, K. Brix, W. Dahmen, Ch. Hohn, S. Mogosan, S. Müller, and G. Schieffer. Parallel and adaptive methods for fluid-structure-interactions. Numerical Notes on Fluid Mechanics, 2009. Submitted. Google Scholar
  5. 5.
    R. Becker and R. Rannacher. A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Num. Math., 4:237–264, 1996. MATHMathSciNetGoogle Scholar
  6. 6.
    R. Becker and R. Rannacher. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer., 10:1–102, 2001. MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    J. Bell, M.J. Berger, J. Saltzman, and M. Welcome. Three-dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Comput., 15(1):127–138, 1994. MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M.J. Berger and P. Colella. Local adaptive mesh refinement for shock hydrodynamics. J. Comp. Physics, 82:64–84, 1989. MATHCrossRefGoogle Scholar
  9. 9.
    M.J. Berger and R.J. LeVeque. Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal., 35(6):2298–2316, 1998. MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M.J. Berger and J. Oliger. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comp. Physics, 53:484–512, 1984. MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    B. Bihari. Multiresolution schemes for conservation laws with viscosity. J. Comp. Phys., 123(1):207–225, 1996. MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    B. Bihari and A. Harten. Multiresolution schemes for the numerical solution of 2–D conservation laws I. SIAM J. Sci. Comput., 18(2):315–354, 1997. MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    F. Bramkamp. Unstructured h -Adaptive Finite-Volume Schemes for Compressible Viscous Fluid Flow. PhD thesis, RWTH Aachen, 2003. http://darwin.bth.rwth-aachen.de/opus3/volltexte/2003/725/03_255.pdf.
  14. 14.
    F. Bramkamp, B. Gottschlich-Müller, M. Hesse, Ph. Lamby, S. Müller, J. Ballmann, K.-H. Brakhage, and W. Dahmen. H-adaptive Multiscale Schemes for the Compressible Navier–Stokes Equations — Polyhedral Discretization, Data Compression and Mesh Generation. In J. Ballmann, editor, Flow Modulation and Fluid-Structure-Interaction at Airplane Wings, volume 84 of Numerical Notes on Fluid Mechanics, pages 125–204. Springer, 2003. Google Scholar
  15. 15.
    F. Bramkamp, Ph. Lamby, and S. Müller. An adaptive multiscale finite volume solver for unsteady an steady state flow computations. J. Comp. Phys., 197(2):460–490, 2004. MATHCrossRefGoogle Scholar
  16. 16.
    A. Brandt. Multi-level adaptive solutions to boundary-value problems. Math. Comp., 31:333–390, 1977. MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    A. Brandt. Multi-level adaptive techniques (mlat) for partial differential equations: Ideas and software. In Mathematical software III, Proc. Symp., Madison 1977, pages 277–318, 1977. Google Scholar
  18. 18.
    K. Brix, S. Mogosan, S. Müller, and G. Schieffer. Parallelization of multiscale-based grid adaptation using space-filling curves. IGPM–Report 299, RWTH Aachen, 2009. Google Scholar
  19. 19.
    R. Bürger, R. Ruiz, and K. Schneider. Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux. J. Eng. Math., 60(3-4):365–385, 2008. MATHCrossRefGoogle Scholar
  20. 20.
    R. Bürger, R. Ruiz, K. Schneider, and M.A. Sepulveda. Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension. ESAIM, Math. Model. Numer. Anal., 42(4):535–563, 2008. MATHCrossRefGoogle Scholar
  21. 21.
    J.M. Carnicer, W. Dahmen, and J.M. Peña. Local decomposition of refinable spaces and wavelets. Appl. Comput. Harmon. Anal., 3:127–153, 1996. MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet schemes for nonlinear variational schemes. Numer. Anal., 41(5):1785–1823, 2003. MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    A. Cohen, W. Dahmen, and R. DeVore. Sparse evaluation of compositions of functions using multiscale expansions. SIAM J. Math. Anal., 35(2):279–303, 2003. MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    A. Cohen, I. Daubechies, and J. Feauveau. Bi–orthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math., 45:485–560, 1992. MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    A. Cohen, S.M. Kaber, S. Müller, and M. Postel. Fully Adaptive Multiresolution Finite Volume Schemes for Conservation Laws. Math. Comp., 72(241):183–225, 2003. MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    A. Cohen, S.M. Kaber, and M. Postel. Multiresolution Analysis on Triangles: Application to Gas Dynamics. In G. Warnecke and H. Freistühler, editors, Hyperbolic Problems: Theory, Numerics, Applications, pages 257–266. Birkhäuser, 2002. Google Scholar
  27. 27.
    F. Coquel, Q.L. Nguyen, M. Postel, and Q.H. Tran. Local time stepping applied to implicit-explicit methods for hyperbolic systems. SIAM Multiscale Modeling and Simulation, 2009. Accepted for publication. Google Scholar
  28. 28.
    F. Coquel, M. Postel, N. Poussineau, and Q.H. Tran. Multiresolution technique and explicit-implicit scheme for multicomponent flows. J. Numer. Math., 14:187–216, 2006. MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    W. Dahmen and S. Müller. Multiscale techniques for high-resolved vortex strcutures, 2008. DFG project Multiresolution and Adaptive Methods for Convection-Dominated Problems, http://www.sfbtr40.de/index.php?option=com_content&view=category&layout=blog&id=34&Itemid=58&lang=en
  30. 30.
    M. Domingues, O. Roussel, and K. Schneider. On space-time adaptive schemes for the numerical solution of partial differential equations. ESAIM: Proceedings, 16:181–194, 2007. MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    R. Donat. Using Harten’s multiresolution framework on existing high resolution shock capturing schemes, 2009. Presentation at Workshop on Multiresolution and Adaptive Methods for Convection-Dominated Problems, http://www.ann.jussieu.fr/mamcdp09/slides/RosaDonatMAMCDP09.pdf.
  32. 32.
    J. Edwards and M.S. Liou. Low-diffusion flux-splitting methods for flows at all speeds. AIAA Journal, 36:1610–1617, 1998. CrossRefGoogle Scholar
  33. 33.
    K. Eriksson and C. Johnson. Adaptive finite element methods for parabolic problems. IV. Nonlinear problems. SIAM J. Numer. Anal., 32:1729–1749, 1995. MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    M. Farge and K. Schneider. Coherent vortex simulation (cvs), a semi-deterministic turbulence model using wavelets. Turbulence and Combustion, 66(4):393–426, 2001. MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    M. Farge, K. Schneider, and N. Kevlahan. Non-gaussianity and coherent vortex simulation for two-dimensional turbulence using an orthogonal wavelet basis. Phys. Fluids, 11(8):2187–2201, 1999. MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    L. Ferm and P. Lötstedt. Space-time adaptive solution of first order PDEs. J. Sci. Comput., 26(1):83–110, 2006. MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    B. Gottschlich–Müller. Multiscale Schemes for Conservation Laws. PhD thesis, RWTH Aachen, 1998. Google Scholar
  38. 38.
    A. Harten. Discrete multi–resolution analysis and generalized wavelets. J. Appl. Num. Math., 12:153–193, 1993. MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    A. Harten. Adaptive multiresolution schemes for shock computations. J. Comp. Phys., 115:319–338, 1994. MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    A. Harten. Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math., 48(12):1305–1342, 1995. MATHMathSciNetGoogle Scholar
  41. 41.
    A. Harten. Multiresolution representation of data: A general framework. SIAM J. Numer. Anal., 33(3):1205–1256, 1996. MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    R. Hartmann and R. Rannacher. Adaptive FE-methods for conservation laws. In G. Warnecke and H. Freistühler, editors, Hyperbolic Problems: Theory, Numerics, Applications, pages 495–504. Birkhäuser, 2002. Google Scholar
  43. 43.
    P. Houston, J.A. Mackenzie, E. Süli, and G. Warnecke. A posteriori error analysis for numerical approximations of Friedrichs systems. Numer. Math., 82:433–470, 1999. MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    N. Hovhannisyan and S. Müller. On the stability of fully adaptive multiscale schemes for conservation laws using approximate flux and source reconstruction strategies. IGPM–Report 284, RWTH Aachen, 2008. Accepted for publication in IMA Journal of Numerical Analysis. Google Scholar
  45. 45.
    J. Jeong and F. Hussain. On the identification of a vortex. Journal of Fluid Mechanics, 285:69–94, 1995. MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    G. Karypis and V. Kumar. Multilevel algorithms for multi-constraint graph partitioning. Supercomputing, 1998. Google Scholar
  47. 47.
    G. Karypis and V. Kumar. A parallel algorithm for multilevel graph partitioning and sparse matrix ordering. Journal of Parallel and Distributed Computing, 48:71–85, 1998. CrossRefGoogle Scholar
  48. 48.
    D. Kröner and M. Ohlberger. A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions. Math. Comp., 69(229):25–39, 2000. MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    S.N. Kruzhkov. First order quasilinear equations with several space variables. Math. USSR Sb., 10:217–243, 1970. MATHCrossRefGoogle Scholar
  50. 50.
    N.N. Kuznetsov. The weak solution of the Cauchy problem for a multi–dimensional quasilinear equation. Mat. Zametki, 2:401–410, 1967. In Russian. MATHMathSciNetGoogle Scholar
  51. 51.
    Ph. Lamby. Parametric Multi-Block Grid Generation and Application To Adaptive Flow Simulations. PhD thesis, RWTH Aachen, 2007. http://darwin.bth.rwth-aachen.de/opus3/volltexte/2007/1999/pdf/Lamby_Philipp.pdf.
  52. 52.
    Ph. Lamby, R. Massjung, S. Müller, and Y. Stiriba. Inviscid flow on moving grids with multiscale space and time adaptivity. In Numerical Mathematics and Advanced Applications: Proceedings of Enumath 2005 the 6th European Conference on Numerical Mathematics and Advanced Mathematics, pages 755–764. Springer, 2006. Google Scholar
  53. 53.
    Ph. Lamby, S. Müller, and Y. Stiriba. Solution of shallow water equations using fully adaptive multiscale schemes. Int. Journal for Numerical Methods in Fluids, 49(4):417–437, 2005. MATHCrossRefGoogle Scholar
  54. 54.
    S. Müller. Adaptive Multiscale Schemes for Conservation Laws, volume 27 of Lecture Notes on Computational Science and Engineering. Springer, 2002. Google Scholar
  55. 55.
    S. Müller, Ph. Helluy, and J. Ballmann. Numerical simulation of a single bubble by compressible two-phase fluids. Int. Journal for Numerical Methods in Fluids, 2009. DOI 10.1002/fld.2033.
  56. 56.
    S. Müller and Y. Stiriba. Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. Journal for Scientific Computing, 30(3):493–531, 2007. MATHCrossRefGoogle Scholar
  57. 57.
    S. Müller and Y. Stiriba. A multilevel finite volume method with multiscale-based grid adaptation for steady compressible flow. Journal of Computational and Applied Mathematics, Special Issue: Emergent Applications of Fractals and Wavelets in Biology and Biomedicine, 227(2):223–233, 2009. DOI 10.1016/j.cam.2008.03.035. MATHGoogle Scholar
  58. 58.
    S. Osher and R. Sanders. Numerical approximations to nonlinear conservation laws with locally varying time and space grids. Math. Comp., 41:321–336, 1983. MATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    S.A. Pandya, S. Venkateswaran, and T.H. Pulliam. Implementation of preconditioned dual-time procedures in overflow. AIAA Paper 2003-0072, 2003. Google Scholar
  60. 60.
    J.J. Quirk. An adaptive grid algorithm for computational shock hydrodynamics. PhD thesis, Cranfield Institute of Technology, 1991. Google Scholar
  61. 61.
    O. Roussel and K. Schneider. Adaptive multiresolution method for combustion problems: Application to flame ball-vortex interaction. Computers and Fluids, 34(7):817–831, 2005. MATHCrossRefGoogle Scholar
  62. 62.
    O. Roussel and K. Schneider. Numerical studies of spherical flame structures interacting with adiabatic walls using an adaptive multiresolution scheme. Combust. Theory Modelling, 10(2):273–288, 2006. CrossRefMathSciNetGoogle Scholar
  63. 63.
    O. Roussel, K. Schneider, A. Tsigulin, and H. Bockhorn. A conservative fully adaptive multiresolution algorithm for parabolic PDEs. J. Comp. Phys., 188(2):493–523, 2003. MATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    K. Schneider and M. Farge. Numerical simulation of a mixing layer in an adaptive wavelet basis. C.R. Acad. Sci. Paris Série II b, 328:263–269, 2001. Google Scholar
  65. 65.
    W. Schröder. Flow Modulation and Fluid-Structure-Interaction at Airplane Wings II. Numerical Notes on Fluid Mechanics. Springer Verlag, 2009. In preparation. Google Scholar
  66. 66.
    T. Sonar, V. Hannemann, and D. Hempel. Dynamic adaptivity and residual control in unsteady compressible flow computation. Math. and Comp. Modelling, 20:201–213, 1994. MATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    T. Sonar and E. Süli. A dual graph–norm refinement indicator for finite volume approximations of the Euler equations. Numer. Math., 78:619–658, 1998. MATHCrossRefMathSciNetGoogle Scholar
  68. 68.
    Ch. Steiner. Adaptive timestepping for conservation laws via adjoint error representation. PhD thesis, RWTH Aachen, 2008. http://darwin.bth.rwth-aachen.de/opus3/volltexte/2009/2679/.
  69. 69.
    Ch. Steiner, S. Müller, and S. Noelle. Adaptive timestep control for weakly instationary solutions of the Euler equations. IGPM–Report 292, RWTH Aachen, 2009. Google Scholar
  70. 70.
    Ch. Steiner and S. Noelle. On adaptive timestepping for weakly instationary solutions of hyperbolic conservation laws via adjoint error control. Communications in Numerical Methods in Engineering, 2008. Accepted for publication. Google Scholar
  71. 71.
    G. Zumbusch. Parallel multilevel methods. Adaptive mesh refinement and loadbalancing. Advances in Numerical Mathematics. Teubner, Wiesbaden, 2003. Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Institut für Geometrie und Praktische MathematikRWTH Aachen UniversityAachenGermany

Personalised recommendations