×

A spacetime discontinuous Galerkin method for scalar conservation laws. (English) Zbl 1077.65108

The authors introduce a discontinuous Galerkin method for scalar hyperbolic conservation laws. The grid consists of regular polyhedrons in space and time, and it is constructed adaptively as the solution progresses in time. The basis consists of polynomials of degree \(p\) on each grid cell. A slope limiter is used to control overshoots and undershoots at shocks.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adjerid, S.; Flaherty, J. E.; 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] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems, (Oxford Mathematical Monographs (2000), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York) · Zbl 0957.49001
[3] Baumann, C. E.; Oden, J. T., An adaptive-order discontinuous Galerkin method for the solution of the Euler equations of gas dynamics, Int. J. Numer. Methods Engrg., 47, 61-73 (2000) · Zbl 0984.76040
[4] Baumann, C. E.; Oden, J. T., A discontinuous hp finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 175, 311-341 (1999) · Zbl 0924.76051
[5] Bey, K. S.; Oden, J. T., hp-Version discontinuous Galerkin methods for hyperbolic conservation laws, Comput. Methods Appl. Mech. Engrg., 133, 259-286 (1996) · Zbl 0894.76036
[6] Brezzi, F.; Bristeau, M.-O.; Franca, L.; Mallet, M.; Rogé, G., A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. Methods Appl. Mech. Engrg., 96, 117-129 (1992) · Zbl 0756.76044
[7] Brooks, A. N.; Hughes, T. J.R., Streamline-Upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 199-259 (1982) · Zbl 0497.76041
[8] Cockburn, B., Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws, J. Comput. Appl. Math., 128, 187-204 (2001) · Zbl 0974.65092
[9] Cockburn, B.; Gremaud, P.-A., Error estimates for finite element methods for hyperbolic conservation laws, SIAM J. Numer. Anal., 33, 522-554 (1996) · Zbl 0861.65077
[10] Cockburn, B.; Shu, C. W., The Runge-Kutta local projection \(P^1\)-discontinuous-Galerkin finite element method for scalar conservation laws, Math. Modell. Numer. Anal., 25, 337-361 (1991) · Zbl 0732.65094
[11] Cockburn, B.; Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comput., 52, 411-435 (1989) · Zbl 0662.65083
[12] 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, 90-113 (1989) · Zbl 0677.65093
[13] Cockburn, B.; Hou, S.; Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Math. Comput., 54, 545-581 (1990) · Zbl 0695.65066
[14] 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)
[15] Cockburn, B.; Karniadakis, G. E.; Shu, C. W., Discontinuous Galerkin Methods-Theory, Computation and Applications, (Lecture Notes in Computational Science and Engineering, vol. 11 (2000), Springer-Verlag: Springer-Verlag New York)
[16] J. Erickson, D. Guoy, J.M. Sullivan, A. Üngor, Building spacetime meshes over arbitrary spatial domains, Proc. 11th Int. Meshing Roundtable, Ithaca, NY, 2002, pp. 391-402; J. Erickson, D. Guoy, J.M. Sullivan, A. Üngor, Building spacetime meshes over arbitrary spatial domains, Proc. 11th Int. Meshing Roundtable, Ithaca, NY, 2002, pp. 391-402
[17] Harten, A.; Osher, S., Uniformly high-order accurate nonoscillatory schemes, SIAM J. Numer. Anal., 24, 279-309 (1987) · Zbl 0627.65102
[18] Hughes, T. J.R.; Franca, L. P.; Hulbert, G. M., A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive systems, Comput. Methods Appl. Mech. Engrg., 73, 173-189 (1989) · Zbl 0697.76100
[19] Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg., 58, 305-328 (1986) · Zbl 0622.76075
[20] Jaffré, J.; Johnson, C.; Szepessy, A., Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Math. Models Methods Appl. Sci., 5, 367-386 (1995) · Zbl 0834.65089
[21] Johnson, C.; Szepessy, A., On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comput., 49, 427-444 (1987) · Zbl 0634.65075
[22] Johnson, C.; Pitkaranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comput., 46, 1-26 (1986) · Zbl 0618.65105
[23] Kruzkov, S. N., First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81, 228-255 (1970), (in Russian) · Zbl 0202.11203
[24] Lesaint, P.; Raviart, P. A., On a finite element method for solving the neutron transport equation, (DeBoor, C., Mathematical Aspects of Finite Element Methods in Partial Differential Equations (1974), Academic Press: Academic Press NewYork), 89-123
[25] Leveque, R. J., Numerical Methods for Conservation Laws (1992), Birkhauser: Birkhauser Basel · Zbl 0847.65053
[26] R.B. Lowrie, Compact higher-order numerical methods for hyperbolic conservation laws, Ph.D. Dissertation, University of Michigan, 1996; R.B. Lowrie, Compact higher-order numerical methods for hyperbolic conservation laws, Ph.D. Dissertation, University of Michigan, 1996
[27] W.F. Reed, T.R. Hill, Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos National Laboratory, 1973; W.F. Reed, T.R. Hill, Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos National Laboratory, 1973
[28] Richter, G. R., An explicit finite element method for the wave equation, Appl. Numer. Math., 16, 65-80 (1994) · Zbl 0816.65062
[29] Shakib, F.; Hughes, T. J.R.; Johan, A., A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 89, 141-219 (1991) · Zbl 0838.76040
[30] Van Leer, B., Towards the ultimate conservative difference scheme III. Upstream-centered finite difference schemes for ideal compressible flow, J. Comput. Phys., 23, 175-263 (1977) · Zbl 0339.76039
[31] 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.