Computational and Applied Mathematics

, Volume 35, Issue 2, pp 321–349 | Cite as

Adaptive multiresolution discontinuous Galerkin schemes for conservation laws: multi-dimensional case

  • Nils GerhardEmail author
  • Siegfried Müller


The concept of multiresolution-based adaptive DG schemes for non-linear one-dimensional hyperbolic conservation laws has been developed and investigated analytically and numerically in (Math Comp, doi: 10.1090/S0025-5718-2013-02732-9, 2013). The key idea is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids for the data given on a uniformly refined mesh. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. The focus of the present work lies on the extension of the originally one-dimensional concept to higher dimensions and the verification of the choice for the threshold value by means of parameter studies performed for linear and non-linear scalar conservation laws.


Discontinuous Galerkin Grid adaptivity Multiwavelets  Multiresolution analysis Conservation laws 

Mathematics Subject Classification

35L65 65M60 



Financial support has been provided by the German Research Foundation (Deutsche Forschungsgemeinschaft — DFG) in the framework of the Collaborative Research Center SFB-TR-40 and the Research Unit FOR 1779.


  1. Adjerid S, Devine K, Flaherty J, Krivodonova L (2002) A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191:1097–1112MathSciNetCrossRefzbMATHGoogle Scholar
  2. Alpert B (1993) A class of bases in \(l^2\) for the sparse representation of integral operators. SIAM J. Math. Anal. 24:246–262MathSciNetCrossRefzbMATHGoogle Scholar
  3. Alpert B, Beylkin G, Gines D, Vozovoi L (2002) Adaptive solution of partial differential equation in multiwavelet bases. J. Comput. Phys. 182:149–190MathSciNetCrossRefzbMATHGoogle Scholar
  4. Archibald R, Fann G, Shelton W (2011) Adaptive discontinuous Galerkin methods in multiwavelets bases. Appl. Numer. Math. 61:879–890MathSciNetCrossRefzbMATHGoogle Scholar
  5. Arnold DN, Brezzi F, Cockburn B, Marini LD (2002) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5):1749–1779MathSciNetCrossRefzbMATHGoogle Scholar
  6. Barth, T., Jepserson, D.: The design and application of upwind schemes on unstructered meshes. In: AIAA, 27th Aerospace Sciences Meeting (1989)Google Scholar
  7. Bey K, Oden J (1996) \(hp\)-version discontinuous Galerkin methods for hyperbolic conservation laws. Comput. Methods Appl. Mech. Eng. 133(3–4):259–286MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bramkamp F, Lamby P, Müller S (2004) An adaptive multiscale finite volume solver for unsteady and steady state flow computations. J. Comput. Phys. 197(2):460–490MathSciNetCrossRefzbMATHGoogle Scholar
  9. Bürger R, Ruiz-Baier R, Schneider K (2010) Adaptive multiresolution methods for the simulation of waves in excitable media. J. Sci. Comput. 43:261–290MathSciNetCrossRefzbMATHGoogle Scholar
  10. Calle J, Devloo P, Gomes S (2005) Wavelets and adaptive grids for the discontinuous Galerkin method. Numer. Algorithms 39(1–3):143–154MathSciNetCrossRefzbMATHGoogle Scholar
  11. Christov I, Popov B (2008) New non-oscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws. J. Comput. Phys. 227:5736–5757MathSciNetCrossRefzbMATHGoogle Scholar
  12. Cockburn, B., Karniadakis, G., Shu, C.W.: The development of discontinuous Galerkin methods. Cockburn, B., Bernardo (ed.) et al., Discontinuous Galerkin methods. Theory, computation and applications. 1st international symposium on DGM, Newport, RI, USA, May 24–26, 1999. Berlin: Springer. Lect. Notes Comput. Sci. Eng. vol. 11, pp. 3–50 (2000)Google Scholar
  13. Cockburn B, Lin SY, Shu CW (1989) TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84:90–113Google Scholar
  14. Cockburn B, Shu CW (1989) TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52(186):411–435MathSciNetzbMATHGoogle Scholar
  15. Cockburn B, Hou S, Shu CW (1990) The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: the multidimensional case. Math. Comput. 54(190):545–581MathSciNetzbMATHGoogle Scholar
  16. Cockburn B, Shu CW (1998) The Runge–Kutta discontinuous Galerkin method for conservation laws v: multidimensional systems. J. Comput. Phys. 141:199–244MathSciNetCrossRefzbMATHGoogle Scholar
  17. Cockburn B, Shu CW (2001) Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3):173–261MathSciNetCrossRefzbMATHGoogle Scholar
  18. Cohen A (2003) Numerical analysis of wavelet methods. In: Studies in Mathematics and its Applications, vol. 32. North-Holland Publishing Co., AmsterdamGoogle Scholar
  19. Cohen A, Daubechies I, Feauveau JC (1992) Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45:485–560MathSciNetCrossRefzbMATHGoogle Scholar
  20. Coquel F, Maday Y, Müller S, Postel M, Tran Q (2009) New trends in multiresolution and adaptive methods for convection-dominated problems. ESAIM Proc. 29:1–7CrossRefzbMATHGoogle Scholar
  21. Dahmen W, Gottschlich-Müller B, Müller S (2000) Multiresolution schemes for conservation laws. Numer. Math. 88(3):399–443MathSciNetCrossRefzbMATHGoogle Scholar
  22. Dedner A, Makridakis C, Ohlberger M (2007) Error control for a class of Runge Kutta discontinuous Galerkin methods for nonlinear conservation laws. SIAM J. Numer. Anal. 45:514–538MathSciNetCrossRefzbMATHGoogle Scholar
  23. Domingues M, Gomes S, Roussel O, Schneider K (2008) An adaptive multiresolution scheme with local time stepping for evolutionary pdes. J. Comput. Phys. 227:3758–3780MathSciNetCrossRefzbMATHGoogle Scholar
  24. Gerhard N, Iacono F, May G, Müller S, Schäfer R (2014) A high-order discontinuous Galerkin discretization with multiwavelet-based grid adaptation for compressible flows. J. Sci. Comput. doi: 10.1007/s10915-014-9846-9
  25. Godlewski, E., Raviart, P.A.: Hyperbolic systems of conservation laws, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 3/4. Ellipses, Paris (1991)Google Scholar
  26. Gottlieb S, Shu CW, Tadmor E (2001) Strong stability preserving high-order time discretization methods. SIAM Rev. 43(1):89–112MathSciNetCrossRefzbMATHGoogle Scholar
  27. Gottschlich-Müller, B., Müller, S.: Adaptive finite volume schemes for conservation laws based on local multiresolution techniques. In: Hyperbolic problems: theory, numerics, applications. In: Proceedings of the 7th international conference, Zürich, Switzerland, February 1998. Vol. I, pp. 385–394. Birkhäuser, Basel (1999)Google Scholar
  28. Guermond J, Pasquetti R, Popov B (2011) Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230:4248–4267MathSciNetCrossRefzbMATHGoogle Scholar
  29. Harten A (1993) Discrete multi-resolution analysis and generalized wavelets. Appl. Numer. Math. 12:153–192MathSciNetCrossRefzbMATHGoogle Scholar
  30. Harten A (1995) Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Commun. Pure Appl. Math. 48:1305–1342MathSciNetCrossRefzbMATHGoogle Scholar
  31. Hartmann R, Houston P (2002) Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws. SIAM J. Sci. Comput. 24:979–1004MathSciNetCrossRefzbMATHGoogle Scholar
  32. Hartmann R, Houston P (2002) Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations. J. Comput. Phys. 183:508–532MathSciNetCrossRefzbMATHGoogle Scholar
  33. Houston P, Senior B, Süli E (2002) \(hp\)-discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity. Int. J. Numer. Methods Fluids 40(1–2):153–169Google Scholar
  34. Hovhannisyan N, Müller S, Schäfer R (2014) Adaptive multiresolution discontinuous galerkin schemes for conservation laws. Math. Comput. 83(285):113–151Google Scholar
  35. Kaibara, M., Gomes, S.: A fully adaptive multiresolution scheme for shock computations. In: Godunov methods. Theory and applications. International conference, Oxford, GB, October 1999, pp. 497–503. Kluwer Academic/Plenum Publishers, New York (2001)Google Scholar
  36. Keinert, F.: Wavelets and multiwavelets. In: Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton (2004)Google Scholar
  37. Kurganov A, Petrova G, Popov B (2007) Adaptive semidiscrete central-upwind schemes for nonconvex hyperbolic conservation laws. SIAM J. Sci. Comput. 29(6):1064–8275MathSciNetCrossRefzbMATHGoogle Scholar
  38. Mallat SG (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11(7):674–693CrossRefzbMATHGoogle Scholar
  39. Müller, S.: Adaptive multiscale schemes for conservation laws. In: Lecture Notes in Computational Science and Engineering, vol. 27. Springer, Berlin (2003)Google Scholar
  40. Müller, T.: Construction of genuinely multi-dimensional multiwavelets. Bachelor thesis, RWTH Aachen University (2013)Google Scholar
  41. Müller, S.: Multiresolution schemes for conservation laws. DeVore, Ronald (ed.) et al., Multiscale, nonlinear and adaptive approximation. Dedicated to Wolfgang Dahmen on the occasion of his 60th birthday, pp. 379–4080. Springer, Berlin (2009)Google Scholar
  42. Müller S, Helluy P, Ballmann J (2010) Numerical simulation of a single bubble by compressible two-phase fluids. Int. J. Numer. Meth. F. 62(6):591–631MathSciNetzbMATHGoogle Scholar
  43. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory, Report (LA-UR-73-479) (1973)Google Scholar
  44. Remacle JF, Flaherty J, Shephard M (2003) An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Rev. 45(1):53–72MathSciNetCrossRefzbMATHGoogle Scholar
  45. Roussel, O., Schneider, K.: A fully adaptive multiresolution scheme for 3D reaction-diffusion equations. In: Finite volumes for complex applications III. Problems and perspectives. Papers from the 3rd symposium of finite volumes for complex applications, Porquerolles, France, June 24–28, 2002, pp. 833–840. Hermes Penton Science, London (2002)Google Scholar
  46. Schäfer, R.: Adaptive multiresolution discontinuous Galerkin schemes for conservation laws. Ph.D. thesis, RWTH Aachen University (2011)Google Scholar
  47. Schneider K, Roussel O (2010) Coherent vortex simulation of weakly compressible turbulent mixing layers using adaptive multiresolution methods. J. Comput. Phys. 229(6):2267–2286MathSciNetCrossRefzbMATHGoogle Scholar
  48. Schneider K, Vasilyev O (2010) Wavelet methods in computational fluid dynamics. Annu. Rev. Fluid Mech. 42:473–503MathSciNetCrossRefzbMATHGoogle Scholar
  49. Shelton, A.B.: A multi-resolution discontinuous Galerkin method for unsteady compressible flows. Ph.D. thesis, Georgia Institute of Technology (2008)Google Scholar
  50. Strela, V.: Multiwavelets: Theory and applications. Ph.D. thesis, Massachusetts Institute of Technology (1996)Google Scholar
  51. Wang L, Mavriplis D (2009) Adjoint-based \(hp\) adaptive discontinuous Galerkin methods for the 2d compressible Euler equations. J. Comput. Phys. 228(20):7643–7661MathSciNetCrossRefzbMATHGoogle Scholar
  52. Yu, T., Kolarov, K., Lynch, W.: Barysymmetric multiwavelets on triangles. IRC Report 1997–006, Standford University (1997)Google Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2014

Authors and Affiliations

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

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