On source terms and boundary conditions using arbitrary high order discontinuous Galerkin schemes

• Autorzy: Michael Dumbser , Claus-dieter Munz
• Języki publikacji: EN

Abstrakty

EN

This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann problem, originally introduced by Toro and Titarev in a finite volume context, provides simultaneously a numerical flux function as well as a time integration method. The resulting scheme is extremely local since it integrates the PDE from one time step to the successive one in a single step using only information from the direct side neighbors. Since source terms are directly incorporated into the numerical flux via the solution of the GRP, our very high order accurate method is also able to maintain very well smooth steady-state solutions of PDEs with source terms, similar to the so-called well-balanced schemes which are usually specially designed for this purpose. Boundary conditions are imposed solving inverse generalized Riemann problems. Furthermore, we show numerical evidence proving that by using very high order schemes together with high order polynomial representations of curved boundaries, high quality solutions can be obtained on very coarse meshes.

Słowa kluczowe

EN

ADER approach; boundary conditions; unstructured meshes; source terms; discontinuous Galerkin schemes

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2007
• Tom: 17
• Numer: 3
• Strony: 297-310

Daty

wydano

2007

poprawiono

2006-05-10

(nieznana)

2006-12-15

Twórcy

autor

Michael Dumbser

• Institut für Aerodynamik und Gasdynamik, Pfaffenwaldring 21, D-70550 Stuttgart, Germany

autor

Claus-dieter Munz

• Institut für Aerodynamik und Gasdynamik, Pfaffenwaldring 21, D-70550 Stuttgart, Germany

Bibliografia

• 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.
• Ben-Artzi M. and Falcovz J. (1984): A second-order Godunov-type scheme for compressible fluid dynamics. Journal of Computational Physics, Vol.55, pp.1-32.
• Botta N., Klein R., Langenberg S. and Lutzenkirchen S. (2004): Well balanced finite volume methods for nearly hydrostatic flows. Journal of Computational Physics, Vol.196, pp.539-565.
• Bourgeade A., Le Floch 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 Poincare (C) Analyse non lineaire, Vol.6, pp.437-480.
• Cockburn B., Karniadakis G.E. and Shu C.W. (2000): Discontinuous Galerkin Methods. Springer.
• 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.
• 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.
• Dumbser M. (2005): Arbitrary High Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains. Aachen: Shaker Verlag.
• 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. Gutnicand E. Sonnendrucker, Eds.). EMS Publishing House, pp.295-333.
• 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.
• 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.
• Klein R. (1995): Semi-implicit extension of a Godunov-type scheme based on low mach number asymptotics. I: One-dimensional flow. Journal of Computational Physics, Vol.121, pp.213-237.
• 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 Poincare (C) Analyse non lineaire, Vol.5, pp.179-207.
• Le Veque R.J. (1998): Balancing source terms and flux gradients in high resolution Godunov methods. Journal of Computational Physics, Vol.146, pp.346-365.
• Meister A. (1999): Asymptotic single and multiple scale expansions in the low Mach number lim. SIAM Journal on Applied Mathematics, Vol.60, No.1, pp.256-271.
• Meister A. (2003): Asymptotic based preconditioning technique for low mach number flows. Zeschrift fur Angewandte Mathematik und Mechanik (ZAMM), Vol.83, pp.3-25.
• Milholen W.E. (2000): An efficient inverse aerodynamic design method for subsonic flows. Technical Report No.2000-0780, American Institute of Aeronauticsand Astronoutics, Reno, NV.
• Press W.H., Teukolsky S.A., Vetterling W.T. and Flannery B.P. (1996): Numerical Recipes in Fortran 77. Cambridge: Cambridge University Press.
• 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.
• 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.
• Stroud A.H. (1971): Approximate Calculation of Multiple Integrals. Englewood Cliffs, NJ: Prentice-Hall.
• Tarev 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.
• Tarev V.A. and Toro E.F. (2005): ADER schemes for three-dimensionalnonlinear hyperbolic systems. Journal of Computational Physics, Vol.204, pp.715-736.
• Toro E.F. (1999): Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd Ed. Springer.
• Toro E.F., Millington R.C. and Nejad L.A.M (2001): Towards very highorder Godunov schemes, In: Godunov Methods. Theory and Applications (E.F. Toro, Ed.). Kluwer/Plenum Academic Publishers, pp.905-938.
• Toro E.F. and Tarev V.A. (2002): Solution of the generalized Riemannproblem for advection-reaction equations. Proceedings of the Royal Society A, Vol.458, pp.271-281