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On Source Terms and Boundary Conditions Using Arbitrary High Order Discontinuous Galerkin Schemes

International Journal of Applied Mathematics and Computer Science's Cover Image
International Journal of Applied Mathematics and Computer Science
Scientific Computation for Fluid Mechanics and Hyperbolic Systems (special issue), Jan Sokołowski and Eric Sonnendrücker (Eds.)

Bassi F. and Rebay S. (1997): High-order accurate discontinuous finite element solution of the 2D Euler equations.Journal of Computational Physics, Vol. 138, pp. 251-285.10.1006/jcph.1997.5454Search in Google Scholar

Ben-Artzi M. and Falcovitz J. (1984): A second-order Godunovtype scheme for compressible fluid dynamics.Journal of Computational Physics, Vol. 55, pp. 1-32.Search in Google Scholar

Botta N., Klein R., Langenberg S. and Lützenkirchen S. (2004): Well balanced finite volume methods for nearly hydrostatic flows.Journal of Computational Physics, Vol. 196, pp. 539-565.10.1016/j.jcp.2003.11.008Search in Google Scholar

Bourgeade A., LeFloch P., and Raviart P.A. (1989): An asymptotic expansion for the solution of the generalized Riemann problem. Part II: Application to the gas dynamics equations.Annales de l'Institut Henri Poincaré (C) Analyse non linéaire, Vol. 6, pp. 437-480.Search in Google Scholar

Cockburn B., Karniadakis G. E. and Shu C.W. (2000): Discontinuous Galerkin Methods. Springer.10.1007/978-3-642-59721-3Search in Google Scholar

Cockburn B. and Shu C.W. (1989): TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II: General framework.Mathematics of Computation, Vol. 52, pp. 411-435.Search in Google Scholar

Cockburn B. and Shu C. W. (1998): The Runge-Kutta discontinuous Galerkin method for conservation laws. V: Multidimensional systems.Journal of Computational Physics, Vol. 141, pp. 199-224.10.1006/jcph.1998.5892Search in Google Scholar

Dumbser M. (2005): Arbitrary High Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains. Aachen: Shaker Verlag.Search in Google Scholar

Dumbser M. and Munz C.D. (2005): Arbitrary high order Discontinuous Galerkin schemes, In: Numerical Methods for Hyperbolic and Kinetic Problems (S. Cordier, T. Goudon, M. Gutnic and E. Sonnendrucker, Eds.). EMS Publishing House, pp. 295-333.Search in Google Scholar

Dumbser M. and Munz C.D. (2006): Building blocks for arbitrary high order discontinuous Galerkin schemes.Journal of Scientific Computing, Vol. 27, pp. 215-230.10.1007/s10915-005-9025-0Search in Google Scholar

Greenberg J.M. and Le Roux A.Y. (1996): A well-balanced scheme for the numerical processing of source terms in hyperbolic equations.SIAM Journal on Numerical Analysis, Vol. 33, pp. 1-16.10.1137/0733001Search in Google Scholar

Klein R. (1995): Semi-implicit extension of a Godunovtype scheme based on low mach number asymptotics. I: One-dimensional flow.Journal of Computational Physics, Vol. 121, pp. 213-237.Search in Google Scholar

Le Floch P. and Raviart P.A. (1988): An asymptotic expansion for the solution of the generalized riemann problem. Part I: General theory.Annales de l'Institut Henri Poincaré (C) Analyse non linéaire, Vol. 5, pp. 179-207.Search in Google Scholar

LeVeque R.J. (1998): Balancing source terms and flux gradients in high resolution Godunov methods.Journal of Computational Physics, Vol. 146, pp. 346-365.10.1006/jcph.1998.6058Search in Google Scholar

Meister A. (1999): Asymptotic single and multiple scale expansions in the low Mach number limit.SIAM Journal on Applied Mathematics, Vol. 60, No. 1, pp. 256-271.10.1137/S0036139998343198Search in Google Scholar

Meister A. (2003): Asymptotic based preconditioning technique for low mach number flows.Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), Vol. 83, pp. 3-25.10.1002/zamm.200310002Search in Google Scholar

Milholen W.E. (2000): An efficient inverse aerodynamic design method for subsonic flows. Technical Report No. 2000-0780, American Institute of Aeronautics and Astronoutics, Reno, NV.10.2514/6.2000-780Search in Google Scholar

Press W.H., Teukolsky S.A., Vetterling W.T. and Flannery B.P. (1996): Numerical Recipes in Fortran 77. Cambridge: Cambridge University Press.Search in Google Scholar

Qiu J., Dumbser M. and Shu C.W. (2005): The discontinuous Galerkin method with Lax-Wendroff type time discretizations.Computer Methods in Applied Mechanics and Engineering, Vol. 194, pp. 4528-4543.10.1016/j.cma.2004.11.007Search in Google Scholar

Roller S. and Munz C.D. (2000): A low mach number scheme based on multi-scale asymptotics.Computing and Visualization in Science, Vol. 3, pp. 85-91.10.1007/s007910050055Search in Google Scholar

Stroud A.H. (1971): Approximate Calculation of Multiple Integrals. Englewood Cliffs, NJ: Prentice-Hall.Search in Google Scholar

Titarev V.A. and Toro E.F. (2002): ADER: Arbitrary high order Godunov approach.Journal of Scientific Computing, Vol. 17, No. 1-4, pp. 609-618.10.1023/A:1015126814947Search in Google Scholar

Titarev V.A. and Toro E.F. (2005): ADER schemes for three-dimensional nonlinear hyperbolic systems.Journal of Computational Physics, Vol. 204, pp. 715-736.10.1016/j.jcp.2004.10.028Search in Google Scholar

Toro E.F. (1999): Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd Ed. Springer.10.1007/978-3-662-03915-1Search in Google Scholar

Toro E.F., Millington R.C. and Nejad L.A.M (2001): Towards very high order Godunov schemes, In: Godunov Methods. Theory and Applications (E.F. Toro, Ed.). Kluwer/Plenum Academic Publishers, pp. 905-938.Search in Google Scholar

Toro E.F. and Titarev V.A. (2002): Solution of the generalized Riemann problem for advection-reaction equations.Proceedings of the Royal Society A, Vol. 458, pp. 271-281.10.1098/rspa.2001.0926Search in Google Scholar

ISSN:
1641-876X
Language:
English
Publication timeframe:
4 times per year
Journal Subjects:
Mathematics, Applied Mathematics