Communications in Applied Mathematics and Computational Science

Discontinuous Galerkin method with the spectral deferred correction time-integration scheme and a modified moment limiter for adaptive grids

Leandro Gryngarten, Andrew Smith, and Suresh Menon

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The discontinuous Galerkin (DG) method is combined with the spectral deferred correction (SDC) time integration approach to solve the fluid dynamic equations. The moment limiter is generalized for nonuniform grids with hanging nodes that result from adaptive mesh refinement. The effect of characteristic, primitive, or conservative decomposition in the limiting stage is studied. In general, primitive variable decomposition is a better option, especially in two and three dimensions. The accuracy-preserving total variation diminishing (AP-TVD) marker for troubled-cell detection, which uses an averaged-derivative basis, is modified to use the Legendre polynomial basis. Given that the latest basis is generally used for DG, the new approach avoids transforming to the averaged-derivative basis, what results in a more efficient technique. Further, a new error estimator is proposed to determine where to refine or coarsen the grid. This estimator is compared against other estimator used in the literature and shows an improved performance. Canonical tests in one, two, and three dimensions are conducted to show the accuracy of the solver.

Article information

Commun. Appl. Math. Comput. Sci., Volume 7, Number 2 (2012), 133-174.

Received: 15 December 2010
Revised: 8 April 2012
Accepted: 16 April 2012
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 35L65: Conservation laws 35L67: Shocks and singularities [See also 58Kxx, 76L05] 65L06: Multistep, Runge-Kutta and extrapolation methods 65M50: Mesh generation and refinement 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

discontinuous Galerkin moment limiter high-order accuracy adaptive mesh troubled-cell detector spectral deferred correction


Gryngarten, Leandro; Smith, Andrew; Menon, Suresh. Discontinuous Galerkin method with the spectral deferred correction time-integration scheme and a modified moment limiter for adaptive grids. Commun. Appl. Math. Comput. Sci. 7 (2012), no. 2, 133--174. doi:10.2140/camcos.2012.7.133.

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  • G. E. Barter and D. L. Darmofal, Shock capturing with higher-order, PDE-based artificial viscosity, 18th AIAA Computational Fluid Dynamics Conference \normalfont(2007-3823), AIAA, 2007.
  • R. Biswas, K. D. Devine, and J. E. Flaherty, Parallel, adaptive finite element methods for conservation laws, Appl. Numer. Math. 14 (1994), no. 1-3, 255–283.
  • 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), no. 190, 545–581.
  • B. Cockburn, S. Y. Lin, and C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws, III: One-dimensional systems, J. Comput. Phys. 84 (1989), no. 1, 90–113.
  • 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), no. 186, 411–435.
  • ––––, The Runge–Kutta discontinuous Galerkin method for conservation laws, V: Multidimensional systems, J. Comput. Phys. 141 (1998), no. 2, 199–224.
  • ––––, Runge–Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput. 16 (2001), no. 3, 173–261.
  • A. Dutt, L. Greengard, and V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT 40 (2000), no. 2, 241–266.
  • J. E. Flaherty, L. Krivodonova, J.-F. Remacle, and M. S. Shephard, Aspects of discontinuous Galerkin methods for hyperbolic conservation laws, Finite Elem. Anal. Des. 38 (2002), no. 10, 889–908.
  • F. X. Giraldo, J. S. Hesthaven, and T. Warburton, Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations, J. Comput. Phys. 181 (2002), no. 2, 499–525.
  • S. Gottlieb, D. I. Ketcheson, and C.-W. Shu, High order strong stability preserving time discretizations, J. Sci. Comput. 38 (2009), no. 3, 251–289.
  • S. Gottlieb and C.-W. Shu, Total variation diminishing Runge–Kutta schemes, Math. Comp. 67 (1998), no. 221, 73–85.
  • S. Gottlieb, C.-W. Shu, and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001), no. 1, 89–112.
  • J. Grooss and J. S. Hesthaven, A level set discontinuous Galerkin method for free surface flows, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 25-28, 3406–3429.
  • A. Harten, ENO schemes with subcell resolution, J. Comput. Phys. 83 (1989), no. 1, 148–184.
  • J. Jaffré, C. Johnson, and A. Szepessy, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Math. Models Methods Appl. Sci. 5 (1995), no. 3, 367–386.
  • B. Jeon, J. D. Kress, L. A. Collins, and N. Grønbech-Jensen, Parallel tree code for two-component ultracold plasma analysis, Comput. Phys. Commun. 178 (2008), 272–279.
  • G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996), no. 1, 202–228.
  • A. M. Khokhlov, Fully threaded tree algorithms for adaptive refinement fluid dynamics simulations, J. Comput. Phys. 143 (1998), no. 2, 519–543.
  • R. M. Kirby and G. E. Karniadakis, De-aliasing on non-uniform grids: algorithms and applications, J. Comput. Phys. 191 (2003), 249–264.
  • L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, and J. E. Flaherty, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math. 48 (2004), no. 3-4, 323–338.
  • L. Krivodonova, Limiters for high-order discontinuous Galerkin methods, J. Comput. Phys. 226 (2007), no. 1, 879–896.
  • T. Leicht and R. Hartmann, Error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations, J. Comput. Phys. 229 (2010), no. 19, 7344–7360. 2011k:76054
  • Y. Liu, C.-W. Shu, E. Tadmor, and M. Zhang, Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction, SIAM J. Numer. Anal. 45 (2007), no. 6, 2442–2467.
  • Y. Liu, C.-W. Shu, and Z. Xu, Hierarchical reconstruction with up to second degree remainder for solving nonlinear conservation laws, Nonlinearity 22 (2009), no. 12, 2799–2812. 2011c:35338
  • Y. Liu, C.-W. Shu, and M. Zhang, Strong stability preserving property of the deferred correction time discretization, J. Comput. Math. 26 (2008), no. 5, 633–656.
  • M. L. Minion, Semi-implicit spectral deferred correction methods for ordinary differential equations, Commun. Math. Sci. 1 (2003), no. 3, 471–500.
  • P.-O. Persson and J. Peraire, Sub-cell shock capturing for discontinuous Galerkin methods, 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA, January 2006.
  • J. Qiu and C.-W. Shu, A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters, SIAM J. Sci. Comput. 27 (2005), no. 3, 995–1013.
  • ––––, Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method, II: Two dimensional case, Comput. & Fluids 34 (2005), no. 6, 642–663.
  • ––––, Runge–Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Sci. Comput. 26 (2005), no. 3, 907–929.
  • A. Rault, G. Chiavassa, and R. Donat, Shock-vortex interactions at high Mach numbers, J. Sci. Comput. 19 (2003), no. 1-3, 347–371.
  • J.-F. Remacle, J. E. Flaherty, and M. S. Shephard, An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems, SIAM Rev. 45 (2003), no. 1, 53–72.
  • J.-F. Remacle, X. Li, M. S. Shephard, and J. E. Flaherty, Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods, Internat. J. Numer. Methods Engrg. 62 (2005), no. 7, 899–923.
  • C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II, J. Comput. Phys. 83 (1989), no. 1, 32–78.
  • E. F. Toro, Riemann solvers and numerical methods for fluid dynamics: A practical introduction, 2nd ed., Springer, Berlin, 1999.
  • P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys. 54 (1984), no. 1, 115–173.
  • Y. Xia, Y. Xu, and C.-W. Shu, Efficient time discretization for local discontinuous Galerkin methods, Discrete Contin. Dyn. Syst. Ser. B 8 (2007), no. 3, 677–693.
  • J. Xin and J. E. Flaherty, Viscous stabilization of discontinuous Galerkin solutions of hyperbolic conservation laws, Appl. Numer. Math. 56 (2006), no. 3-4, 444–458.
  • Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys. 7 (2010), no. 1, 1–46.
  • Z. Xu, Y. Liu, and C.-W. Shu, Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells, J. Comput. Phys. 228 (2009), no. 6, 2194–2212.
  • M. Yang and Z. J. Wang, A parameter-free generalized moment limiter for high-order methods on unstructured grids, Adv. Appl. Math. Mech. 1 (2009), no. 4, 451–480.
  • L. Yuan and C.-W. Shu, Discontinuous Galerkin method based on non-polynomial approximation spaces, J. Comput. Phys. 218 (2006), no. 1, 295–323.
  • T. Zhou, Y. Li, and C.-W. Shu, Numerical comparison of WENO finite volume and Runge–Kutta discontinuous Galerkin methods, J. Sci. Comput. 16 (2001), no. 2, 145–171.
  • H. Zhu and J. Qiu, Adaptive Runge–Kutta discontinuous Galerkin methods using different indicators: one-dimensional case, J. Comput. Phys. 228 (2009), no. 18, 6957–6976. 2011a:65339