On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes

Journal of Computational Physics

Volumn 229, Issue 23, 2010, Pages 8918-8934

  Zhang, Xiangxiong ^a   Shu, Chi Wang ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

Compressible Euler equations; Discontinuous Galerkin method; Essentially non oscillatory scheme; Finite volume scheme; Gas dynamics; High order accuracy; Hyperbolic conservation laws; Positivity preserving; Weighted essentially non oscillatory scheme

Indexed keywords

COMPRESSIBILITY OF GASES; EULER EQUATIONS; GALERKIN METHODS; GAS DYNAMICS; INCOMPRESSIBLE FLOW; NON NEWTONIAN FLOW;

COMPRESSIBLE EULER EQUATIONS; DISCONTINOUS GALERKIN METHODS; ESSENTIALLY NON-OSCILLATORY SCHEMES; FINITE VOLUME SCHEMES; HIGH-ORDER; HIGHER ORDER ACCURACY; HYPERBOLIC CONSERVATION LAWS; POSITIVITY PRESERVING; WEIGHTED ESSENTIALLY NON-OSCILLATORY SCHEME;

NUMERICAL METHODS;

EID: 77956907524     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.08.016     Document Type: Article

References (28)

1. Cockburn B., Hou S., Shu C.-W. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Mathematics of Computation 1990, 54:545-581.
2. 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. Journal of Computational Physics 1989, 84:90-113.
3. Cockburn B., Shu C.-W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Mathematics of Computation 1989, 52:411-435.
4. Cockburn B., Shu C.-W. The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. Journal of Computational Physics 1998, 141:199-224.
5. Cockburn B., Shu C.-W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing 2001, 16:173-261.
6. Einfeldt B., Munz C.D., Roe P.L., Sjögreen B. On Godunov-type methods near low densities. Journal of Computational Physics 1991, 92:273-295.
7. Gardner C., Dwyer S. Numerical simulation of the XZ Tauri supersonic astrophysical jet. Acta Mathemacita Scientia 2009, 29B:1677-1683.
8. Ha Y., Gardner C. Positive scheme numerical simulation of high Mach number astrophysical jets. Journal of Scientific Computing 2008, 34:247-259.
9. Ha Y., Gardner C., Gelb A., Shu C.-W. Numerical simulation of high Mach number astrophysical jets with radiative cooling. Journal of Scientific Computing 2005, 24:597-612.
10. Harten A. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics 1983, 49:357-393.
11. Harten A., Engquist B., Osher S., Chakravarthy S. Uniformly high order essentially non-oscillatory schemes, III. Journal of Computational Physics 1987, 71:231-303.
12. Harten A., Lax P.D., van Leer B. On upstream differencing and Godunov type schemes for hyperbolic conservation laws. SIAM Review 1983, 25:35-61.
13. Jiang G.-S., Shu C.-W. Efficient implementation of weighted ENO schemes. Journal of Computational Physics 1996, 126:202-228.
14. Korobeinikov V.P. Problems of Point-Blast Theory 1991, American Institute of Physics.
15. T. Linde, P.L. Roe, Robust Euler codes, in: 13th Computational Fluid Dynamics Conference, AIAA Paper-97-2098.
16. Liu W., Cheng J., Shu C.-W. High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations. Journal of Computational Physics 2009, 228:8872-8891.
17. Liu X.-D., Osher S., Chan T. Weighted essentially non-oscillatory schemes. Journal of Computational Physics 1994, 115:200-212.
18. Osher S., Chakravarthy S. High resolution schemes and the entropy condition. SIAM Journal on Numerical Analysis 1984, 21:955-984.
19. Perthame B. Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions. SIAM Journal on Numerical Analysis 1992, 29:1-19.
20. Perthame B., Shu C.-W. On positivity preserving finite volume schemes for Euler equations. Numerische Mathematik 1996, 73:119-130.
21. Sanders R. A third-order accurate variation nonexpansive difference scheme for single nonlinear conservation law. Mathematics of Computation 1988, 51:535-558.
22. Sedov L.I. Similarity and Dimensional Methods in Mechanics 1959, Academic Press, New York.
23. Shu C.-W. TVB uniformly high order schemes for conservation laws. Mathematics of Computation 1987, 49:105-121.
24. Shu C.-W. Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing 1988, 9:1073-1084.
25. Shu C.-W., Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 1988, 77:439-471.
26. Zhang X., Shu C.-W. On maximum-principle-satisfying high order schemes for scalar conservation laws. Journal of Computational Physics 2010, 229:3091-3120.
27. Zhang X., Shu C.-W. A genuinely high order total variation diminishing scheme for one-dimensional scalar conservation laws. SIAM Journal on Numerical Analysis 2010, 48:772-795.
28. X. Zhang, Y. Xia, C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, Journal of Scientific Computing, submitted for publication.