×

zbMATH — the first resource for mathematics

Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. (English) Zbl 1038.65096
Summary: We describe a strategy for detecting discontinuities and for limiting spurious oscillations near such discontinuities when solving hyperbolic systems of conservation laws by high-order discontinuous Galerkin methods. The approach is based on a strong superconvergence at the outflow boundary of each element in smooth regions of the flow. By detecting discontinuities in such variables as density or entropy, limiting may be applied only in these regions; thereby, preserving a high order of accuracy in regions where solutions are smooth. Several one- and two-dimensional flow problems illustrate the performance of these approaches.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adjerid, S.; Devine, K.; Flaherty, J.; Krivodonova, L., A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems, Comput. methods appl. mech. engrg., 191, 1097-1112, (2002) · Zbl 0998.65098
[2] T. Barth, D. Jespersen, The design and application of upwind schemes on unstructured meshes, in: 27th Aerospace Sciences Meeting, Reno, Nevada, AIAA 89-0366, 1989
[3] Biswas, R.; Devine, K.; Flaherty, J., Parallel adaptive finite element methods for conservation laws, Appl. numer. math., 14, 255-284, (1994) · Zbl 0826.65084
[4] Cockburn, B.; Lin, S.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin methods for scalar conservation laws III: one dimensional systems, J. comput. phys., 84, 90-113, (1989) · Zbl 0677.65093
[5] Cockburn, B.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin methods for scalar conservation laws II: general framework, Math. comp., 52, 411-435, (1989) · Zbl 0662.65083
[6] Colella, P.; Glaz, H.M., Efficient solution algorithms for the Riemann problem for real gases, J. comput. phys., 59, 264-289, (1985) · Zbl 0581.76079
[7] V. Dolejsi, M. Feistauer, On the discontinuous Galerkin method for the numerical solution of compressible high-speed flow, Technical Report MATH-knm-2002/1, Charles University, Prague, 2002 · Zbl 1276.76039
[8] Flaherty, J.; Krivodonova, L.; Remacle, J.-F.; Shephard, M., Aspects of discontinuous Galerkin methods for hyperbolic conservation laws, Finite elements anal. design, 38, 889-908, (2002) · Zbl 0996.65106
[9] Goodman, J.; LeVeque, R., A geometric approach to high resolution TVD schemes, SIAM J. numer. anal., 25, 268-284, (1988) · Zbl 0645.65051
[10] Harten, A., High resolution schemes for hyperbolic conservation laws, J. comput. phys., 49, 357-393, (1983) · Zbl 0565.65050
[11] Harten, A.; Hyman, J.; Lax, P., On finite-difference approximations and entropy conditions for shocks, Comm. pure appl. math., 29, 297-322, (1976) · Zbl 0351.76070
[12] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202-228, (1996) · Zbl 0877.65065
[13] Karni, S.; Kurganov, A.; Petrova, G., A smoothness indicator for adaptive algorithms for hyperbolic systems, J. comput. phys., 178, 323-341, (2002) · Zbl 0998.65092
[14] Karniadakis, G.; Sherwin, S., Spectral/hp element methods for CFD, (1999), Oxford University Press New York · Zbl 0954.76001
[15] Krivodonova, L.; Flaherty, J., Error estimation for discontinuous Galerkin solutions of multidimensional hyperbolic problems, Adv. comput. math., 19, 57-71, (2003) · Zbl 1020.65062
[16] Remacle, J.-F.; Flaherty, J.; Shephard, M.S., An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems, SIAM rev., 45, 53-72, (2003) · Zbl 1127.65323
[17] Roe, P., Approximate Riemann solvers, parameter vectors and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066
[18] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. comput. phys., 83, 32-78, (1989) · Zbl 0674.65061
[19] van Leer, B., Towards the ultimate conservation difference scheme, II, J. comput. phys., 14, 361-367, (1974) · Zbl 0276.65055
[20] van Leer, B., Towards the ultimate conservation difference scheme, V, J. comput. phys., 32, 1-136, (1979) · Zbl 1364.65223
[21] Whitham, G., Linear and nonlinear waves, (1974), Wiley New York · Zbl 0373.76001
[22] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.