A generic stabilization approach for higher order discontinuous Galerkin methods for convection dominated problems. (English) Zbl 1229.65175
Summary: We present a stabilized discontinuous Galerkin (DG) method for hyperbolic and convection dominated problems. The presented scheme can be used in several space dimension and with a wide range of grid types. The stabilization method preserves the locality of the DG method and therefore allows to apply the same parallelization techniques used for the underlying DG method. As an example problem we consider the Euler equations of gas dynamics for an ideal gas. We demonstrate the stability and accuracy of our method through the detailed study of several test cases in two space dimension on both unstructured and cartesian grids. We show that our stabilization approach preserves the advantages of the DG method in regions where stabilization is not necessary. Furthermore, we give an outlook to adaptive and parallel calculations in 3d.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
| 35Q31 | Euler equations |
| 76N15 | Gas dynamics (general theory) |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
Keywords:
conservation laws; higher order methods; discontinuous Galerkin; finite volume; generic limiter; numerical examples; convection dominated problems; stabilization; parallelization; Euler equations; gas dynamicsReferences:
| [1] | Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. II: Implementation and tests in Dune. Computing 82(2–3), 121–138 (2008) · Zbl 1151.65088 · doi:10.1007/s00607-008-0004-9 |
| [2] | Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. I: Abstract framework. Computing 82(2–3), 103–119 (2008) · Zbl 1151.65089 · doi:10.1007/s00607-008-0003-x |
| [3] | Burbeau, A., Sagaut, P., Bruneau, Ch.-H.: A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. J. Comput. Phys. 169(1), 111–150 (2001) · Zbl 0979.65081 · doi:10.1006/jcph.2001.6718 |
| [4] | Christov, I., Popov, B.: New non-oscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws. J. Comput. Phys. 227(11), 5736–5757 (2008) · Zbl 1151.65068 · doi:10.1016/j.jcp.2008.02.007 |
| [5] | Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems. J. Comput. Phys. 141, 199–224 (1998) · Zbl 0920.65059 · doi:10.1006/jcph.1998.5892 |
| [6] | Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001) · Zbl 1065.76135 · doi:10.1023/A:1012873910884 |
| [7] | Dedner, A., Klöfkorn, R.: A generic stabilization approach for higher order discontinuous Galerkin methods for convection dominated problems. Preprint No. 8, Universität Freiburg (2008) · Zbl 1229.65175 |
| [8] | Dedner, A., Klöfkorn, R.: Stabilization for discontinuous Galerkin methods applied to systems of conservation laws. In: Proc. of the 12th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, Maryland (2008) · Zbl 1192.35007 |
| [9] | Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M.: A generic interface for parallel and adaptive scientific computing: Abstraction principles and the Dune-Fem module. Preprint No. 3, Mathematisches Institut, Universität Freiburg (2009) · Zbl 1201.65178 |
| [10] | Dedner, A., Makridakis, C., Ohlberger, M.: Error control for a class of Runge-Kutta discontinuous Galerkin methods for nonlinear conservation laws. SIAM J. Numer. Anal. 45(2), 514–538 (2007) · Zbl 1145.65072 · doi:10.1137/050624248 |
| [11] | Dolejší, V., Feistauer, M., Schwab, C.: On some aspects of the discontinuous Galerkin finite element method for conservation laws. Math. Comput. Simul. 61(3–6), 333–346 (2003) · Zbl 1013.65108 · doi:10.1016/S0378-4754(02)00087-3 |
| [12] | Ern, A., Proft, J.: A posteriori discontinuous Galerkin error estimates for transient convection-diffusion equations. Appl. Math. Lett. 18(7), 833–841 (2005) · Zbl 1084.65092 · doi:10.1016/j.aml.2004.05.019 |
| [13] | Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001) · Zbl 0967.65098 · doi:10.1137/S003614450036757X |
| [14] | Hagen, T.R., Lie, K.-A., Natvig, J.R.: Solving the Euler equations on graphics processing units. In: Alexandrov V.N., et al. (eds.) Proceedings, Part IV, Computational Science–ICCS 2006. 6th International Conference, Reading, UK, 28–31 May, 2006. Lecture Notes in Computer Science, vol. 3994, pp. 220–227. Springer, Berlin (2006) · Zbl 1157.76358 |
| [15] | Hoteit, H., Ackerer, Ph., Mosé, R., Erhel, J., Philippe, B.: New two-dimensional slope limiters for discontinuous Galerkin methods on arbitrary meshes. Int. J. Numer. Methods Eng. 61(14), 2566–2593 (2004) · Zbl 1075.76575 · doi:10.1002/nme.1172 |
| [16] | Klieber, W., Rivière, B.: Adaptive simulations of two-phase flow by discontinuous Galerkin methods. Comput. Methods Appl. Mech. Eng. 196(1–3), 404–419 (2006) · Zbl 1120.76327 · doi:10.1016/j.cma.2006.05.007 |
| [17] | Krivodonova, L.: Limiters for high-order discontinuous Galerkin methods. J. Comput. Phys. 226(1), 879–896 (2007) · Zbl 1125.65091 · doi:10.1016/j.jcp.2007.05.011 |
| [18] | Krivodonova, L., Xin, J., Remacle, J.-F., Chevaugeon, N., Flaherty, J.E.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48(3–4), 323–338 (2004) · Zbl 1038.65096 · doi:10.1016/j.apnum.2003.11.002 |
| [19] | Kröner, D.: Numerical Schemes for Conservation Laws. Wiley, Teubner, New York, Leipzig (1997) · Zbl 0872.76001 |
| [20] | Kröner, D., Ohlberger, M.: A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions. Math. Comput. 69, 25–39 (2000) · Zbl 0934.65102 |
| [21] | Luo, H., Baum, J.D., Löhner, R.: A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J. Comput. Phys. 225(1), 686–713 (2007) · Zbl 1122.65089 · doi:10.1016/j.jcp.2006.12.017 |
| [22] | Persson, P.-O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin methods. In: 44th AIAA Aerospace Sciences Meeting, AIAA-2006-0112, Reno, Nevada, Department of Aeronautics & Astronautics. Massachusetts Institute of Technology (2006) |
| [23] | Qiu, J., Shu, C.-W.: A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters. SIAM J. Sci. Comput. 27(3), 995–1013 (2005) · Zbl 1092.65084 · doi:10.1137/04061372X |
| [24] | Qiu, J., Shu, C.-W.: A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters. SIAM J. Sci. Comput. 27(3), 995–1013 (2005) · Zbl 1092.65084 · doi:10.1137/04061372X |
| [25] | Qiu, J., Shu, C.-W.: Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput. 26(3), 907–929 (2005) · Zbl 1077.65109 · doi:10.1137/S1064827503425298 |
| [26] | Wesenberg, M.: Efficient finite–volume schemes for magnetohydrodynamics simulations in solar physics. Ph.D. thesis, Universität Freiburg (2003) · Zbl 1030.76002 |
| [27] | Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984) · Zbl 0573.76057 · doi:10.1016/0021-9991(84)90142-6 |
| [28] | Xin, Jianguo, Flaherty, Joseph E.: Viscous stabilization of discontinuous Galerkin solutions of hyperbolic conservation laws. Appl. Numer. Math. 56(3–4), 444–458 (2006) · Zbl 1089.65101 · doi:10.1016/j.apnum.2005.08.001 |
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