Journal of Scientific Computing

, Volume 66, Issue 1, pp 406–434 | Cite as

A Comparison of Artificial Viscosity, Limiters, and Filters, for High Order Discontinuous Galerkin Solutions in Nonlinear Settings

  • C. Michoski
  • C. Dawson
  • E. J. Kubatko
  • D. Wirasaet
  • S. Brus
  • J. J. Westerink


Nonlinear systems of equations demonstrate complicated regularity features that are often obfuscated by overly diffuse numerical methods. Using a discontinuous Galerkin finite element method, we study a nonlinear system of advection–diffusion–reaction equations and aspects of its regularity. For numerical regularization, we present a family of solutions consisting of: (1) a sharp, computationally efficient slope limiter, known as the BDS limiter, (2) a standard spectral filter, and (3) a novel artificial diffusion algorithm with a solution-dependent entropy sensor. We analyze these three numerical regularization methods on a classical test in order to test the strengths and weaknesses of each, and then benchmark the methods against a large application model.


Discontinuous Galerkin Nonlinear system High order Regularization Slope limiting Spectral filters  Artificial diffusion Artificial viscosity Advection  Diffusion Reaction 



The authors would like to thank Kyle Mandli and Stewart Stafford for helpful comments, insights, and conversation, and to acknowledge the support of the National Science Foundation Grant NSF ACI-1339801.


  1. 1.
    Abgrall, R.: On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys. 114(1), 45–58 (1994). doi: 10.1006/jcph.1994.1148 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/02). ISSN 0036–1429. doi: 10.1137/S0036142901384162
  3. 3.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.: Discontinuous Galerkin methods for elliptic problems. In: Discontinuous Galerkin methods (Newport, RI, 1999). Lecture Notes in Computer Science Engineering, vol. 11, pp. 89–101. Springer, Berlin (2000)Google Scholar
  4. 4.
    Barter, G.E., Darmofal, D.L.: Shock capturing with PDE-based artificial viscosity for DGFEM: part I. Formulation. J. Comput. Phys. 229(5), 1810–1827 (2010). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barth, T., Jesperson, D.C.: The design and application of upwind schems and unstructured meshes. AIAA, paper 89–0366 (1989)Google Scholar
  6. 6.
    Bell, J.B., Dawson, C.N., Shubin, G.R.: An unsplit, higher order godunov method for scalar conservation laws in multiple dimensions. J. Comput. Phys. 74(1), 1–24 (1988). ISSN 0021–9991. doi: 10.1016/0021-9991(88)90065-4.
  7. 7.
    Bunya, S., Kubatko, E.J., Westerink, J.J., Dawson, C.: A wetting and drying treatment for the Runge–Kutta discontinuous Galerkin solution to the shallow water equations. Comput. Methods Appl. Mech. Eng. 198(17–20), 1548–1562 (2009). doi: 10.1016/j.cma.2009.01.008 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bunya, S., Dietrich, J.C., Westerink, J.J., Ebersole, B.A., Smith, J.M., Atkinson, J.H., Jensen, R., Resio, D.T., Luettich, R.A., Dawson, C., Cardone, V.J., Cox, A.T., Powell, M.D., Westerink, H.J., Roberts, H.J.: A high-resolution coupled riverine flow, tide, wind, wind wave, and storm surge model for southern Louisiana and Mississippi. Part I: Model Development and Validation. Mon. Weather Rev. 138(2), 345–377 (2010). doi: 10.1175/2009MWR2906.1 CrossRefGoogle Scholar
  9. 9.
    Casoni, E., Peraire, J., Huerta, A.: One-dimensional shock-capturing for high-order discontinuous Galerkin methods. Int. J. Numer. Methods Fluids 71(6), 737–755 (2013). doi: 10.1002/fld.3682 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cockburn, B.: An introduction to the discontinuous Galerkin method for convection-dominated problems. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Cetraro, 1997). Lecture Notes in Mathematics, vol. 1697, pp. 151–268. Springer, Berlin (1998)Google Scholar
  11. 11.
    Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comp. 52(186), 411–435 (1989)MathSciNetMATHGoogle Scholar
  12. 12.
    Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998). (electronic)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cockburn, B., Shu, C.W.: The runge-kutta discontinuous galerkin method for conservation laws v: multidimensional systems. J. Comput. Phys. 141(2), 199–224(1998). ISSN 0021–9991. doi: 10.1006/jcph.1998.5892.
  15. 15.
    Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dawson, C., Kubatko, E.J., Westerink, J.J., Trahan, C., Mirabito, C., Michoski, C., Panda, N.: Discontinuous Galerkin methods for modeling hurricane storm surge. Adv. Water Resour. 34(9), 1165–1176 (2011). ISSN 0309–1708. doi: 10.1016/j.advwatres.2010.11.004. New Computational Methods and Software Tools
  17. 17.
    Dawson, C., Westerink, J.J., Feyen, J.C., Pothina, D.: Continuous, discontinuous and coupled discontinuous–continuous Galerkin finite element methods for the shallow water equations. Int. J. Numer. Methods Fluids 52(1), 63–88 (2006). doi: 10.1002/fjd.1156 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    DuChene, M., Spagnuolo, A.M., Kubatko, E., Westerink, J., Dawson, C.: A framework for running the ADCIRC discontinuous Galerkin storm surge model on a GPU. Procedia Comput. Sci. 4, 2017–2026 (2011). doi: 10.1016/j.procs.2011.04.220.
  19. 19.
    Durlofsky, L.J., Engquist, B., Osher, S.: Triangle based adaptive stencils for the solution of hyperbolic conservation laws. J. Comput. Phys. 98(1), 64–73 (1992). ISSN 0021–9991. doi: 10.1016/0021-9991(92)90173-V.
  20. 20.
    Feistauer, M., Felcman, J., Straškraba, I.: Mathematical and Computational Methods for Compressible Flow. Numerical Mathematics and Scientific Computation. Oxford University Press (2003). ISBN 0-19-850588-4Google Scholar
  21. 21.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods. In: Texts in Applied Mathematics, vol. 54. Springer, New York, (2008). ISBN 978-0-387-72065-4. doi: 10.1007/978-0-387-72067-8. Algorithms, analysis, and applications
  22. 22.
    Hindenlang, F., Gassner, G.J., Altmann, C., Beck, A., Staudenmaier, M., Munz, C.-D.: Explicit discontinuous Galerkin methods for unsteady problems. Comput. Fluids 61(SI), 86–93 (2012). doi: 10.1016/j.compfluid.2012.03.006 MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ketcheson, D., Parsani, M., LeVeque, R.: High-order wave propagation algorithms for hyperbolic systems. SIAM J. Sci. Comput. 35(1), A351–A377 (2013). doi: 10.1137/110830320 MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kloeckner, A., Warburton, T., Bridge, J., Hesthaven, J.S.: Nodal discontinuous Galerkin methods on graphics processors. J. Comput. Phys. 228(21), 7863–7882 (2009). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Kloeckner, A., Warburton, T., Hesthaven, J.S.: Viscous shock capturing in a time-explicit discontinuous Galerkin method. Math. Model. Nat. Phenom. 6(3), 57–83 (2011). doi: 10.1051/mmnp/20116303 MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kubatko, E.J., Westerink, J.J., Dawson, C.: hp discontinuous galerkin methods for advection dominated problems in shallow water flow. Comput. Methods Appl. Mech. Eng. 196(1–3), 437–451 (2006). ISSN 0045–7825. doi: 10.1016/j.cma.2006.05.002.
  27. 27.
    Kubatko, E.J., Westerink, J.J., Dawson, C.: Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge–Kutta time discretizations. J. Comput. Phys. 222(2), 832–848 (2007a). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kubatko, E.J., Dawson, C., Westerink, J.J.: Time step restrictions for Runge–Kutta discontinuous Galerkin methods on triangular grids. J. Comput. Phys. 227(23), 9697–9710 (2008). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Kubatko, E.J., Bunya, S., Dawson, C., Westerink, J.J., Mirabito, C.: A performance comparison of continuous and discontinuous finite element shallow water models. J. Sci. Comput. 40(1–3), 315–339 (2009). doi: 10.1007/s10915-009-9268-2 MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kubatko, Ethan J., Westerink, Joannes J., Dawson, Clint: Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge–Kutta time discretizations. J. Comput. Phys. 222(2), 832–848 (2007b). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Kubatko, Ethan J., Westerink, Joannes J., Dawson, Clint: hp discontinuous Galerkin methods for advection dominated problems in shallow water flow. Comput. Methods Appl. Mech. Eng. 196(1–3), 437–451 (2006). doi: 10.1016/j.cma.2006.05.002 CrossRefMATHGoogle Scholar
  32. 32.
    Kuzmin, D.: A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods. J. Comput. Appl. Math. 233(12), 3077–3085 (2010). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kuzmin, D.: Slope limiting for discontinuous Galerkin approximations with a possibly non-orthogonal Taylor basis. Int. J. Numer. Methods Fluids 71(9), 1178–1190 (2013). doi: 10.1002/fld.3707 MathSciNetCrossRefGoogle Scholar
  34. 34.
    Kuzmin, D., Schieweck, F.: A parameter-free smoothness indicator for high-resolution finite element schemes. Central Eur. J. Math. 11(8), 1478–1488 (2013). doi: 10.2478/s11533-013-0254-4 MathSciNetMATHGoogle Scholar
  35. 35.
    Liu, Y., Shu, C.-W., Tadmor, E., Zhang, M.: Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction. SIAM J. Numer. Anal. 45(6), 2442–2467 (2007). doi: 10.1137/060666974. (electronic)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Maday, Y., Kaber, S., Tadmor, E.: Legendre pseudospectral viscosity method for nonlinear conservation-laws. SIAM J. Numer. Anal. 30(2), 321–342 (1993). doi: 10.1137/0730016 MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Meister, A., Ortleb, S., Sonar, Th: Application of spectral filtering to discontinuous Galerkin methods on triangulations. Numer. Methods Partial Differ. Equ. 28(6), 1840–1868 (2012). doi: 10.1002/num.20705 MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Michoski, C., Evans, J.A., Schmitz, P.G., Vasseur, A.: A discontinuous Galerkin method for viscous compressible multifluids. J. Comput. Phys. 229(6), 2249–2266 (2010). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Michoski, C., Mirabito, C., Dawson, C., Wirasaet, D., Kubatko, E.J., Westerink, J.J.: Adaptive hierarchic transformations for dynamically \(p\)-enriched slope-limiting over discontinuous Galerkin systems of generalized equations. J. Comput. Phys. 230(22), 8028–8056 (2011). doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Michoski, C., Mirabito, C., Dawson, C., Wirasaet, D., Kubatko, E.J., Westerink, J.J.: Dynamic p-enrichment schemes for multicomponent reactive flows. Adv. Water Resour. 34(12), 1666–1680 (2011). ISSN 0309–1708. doi: 10.1016/j.advwatres.2011.09.001.
  41. 41.
    Michoski, C., Evans, J.A., Schmitz, P.G.: Discontinuous Galerkin hp-adaptive methods for multiscale chemical reactors i: Quiescent reactors. Preprint (2013)Google Scholar
  42. 42.
    Osher, S.: Convergence of generalized MUSCL schemes. SIAM J. Numer. Anal. 22(5), 947–961 (1985)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Persson, P.-O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin methods. Preprint (2013)Google Scholar
  44. 44.
    Ruuth, S.J.: Global optimization of explicit strong-stability-preserving Runge–Kutta methods. Math. Comput. 75(253), 183–207 (2006). doi: 10.1090/S0025-5718-05-01772-2. (electronic)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Tadmor, Eitan, Waagan, Knut: Adaptive spectral viscosity for hyperbolic conservation laws. SIAM J. Sci. Comput. 34(2), A993–A1009 (2012). doi: 10.1137/110836456 MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Thomas, J.W.: Numerical partial differential equations: finite difference methods. In: Texts in Applied Mathematics, vol. 22. Springer, New York (1995). ISBN 0-387-97999-9Google Scholar
  48. 48.
    Wirasaet, D., Tanaka, S., Kubatko, E.J., Westerink, J.J., Dawson, C.: A performance comparison of nodal discontinuous Galerkin methods on triangles and quadrilaterals. Int. J. Numer. Methods Fluids 64(10–12), 1336–1362 (2010). ISSN 0271–2091. doi: 10.1002/fld.2376. 15th International Conference on Finite Elements in Flow Problems, Tokyo, Japan, April 01–03, 2009
  49. 49.
    Xin, J.G., Flaherty, J.E.: Viscous stabilization of discontinuous Galerkin solutions of hyperbolic conservation laws. Appl. Numer. Math. 56(3–4), 444–458 (2006). ISSN 0168–9274. doi: 10.1016/j.apnum.2005.08.001. 3rd International Conference on Numerical Solutions of Volterra and Delay Equations, Tempe, AZ, May, 2004
  50. 50.
    Xing, Y., Shu, C.W.: High-order finite volume weno schemes for the shallow water equations with dry states. Adv. Water Resour. 34(8), 1026–1038 (2011). ISSN 0309–1708. doi: 10.1016/j.advwatres.2011.05.008.
  51. 51.
    Xing, Y., Zhang, X., Shu, C.-W.: Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33(12), 1476–1493 (2010). ISSN 0309–1708. doi: 10.1016/j.advwatres.2010.08.005.
  52. 52.
    Zingan, V., Guermond, J.-L., Morel, J., Popov, B.: Implementation of the entropy viscosity method with the discontinuous Galerkin method. Comput. Methods Appl. Mech. Eng. 253, 479–490 (2013). doi: 10.1016/j.cma.2012.08.018 MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Zingan, V.N.: Discontinuous Galerkin finite element method for the nonlinear hyperbolic problems with entropy-based artificial viscosity stabilization. Dissertation, Texas A&M University (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • C. Michoski
    • 1
  • C. Dawson
    • 1
  • E. J. Kubatko
    • 2
  • D. Wirasaet
    • 3
  • S. Brus
    • 3
  • J. J. Westerink
    • 3
  1. 1.Institute for Computational Engineering and Sciences (ICES), Computational Hydraulics Group (CHG)University of Texas at AustinAustinUSA
  2. 2.Department of Civil and Environmental Enineering and Geodetic ScienceThe Ohio State UniversityColumbusUSA
  3. 3.Computational Hydraulics Laboratory, Department of Civil Engineering and Geological SciencesUniversity of Notre DameNotre DameUSA

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