Residual distribution schemes for conservation laws via adaptive quadrature

SIAM Journal on Scientific Computing

Volumn 24, Issue 3, 2003, Pages 732-769

  Abgrall, Rémi ^a   Barth, Timothy ^b  

^a UNIVERSITÉ BORDEAUX 1   (France)

^b NASA AMES RESEARCH CENTER   (United States)

Author keywords

Entropy symmetrization; Fluctuation splitting; Residual distribution; Symmetric hyperbolic

Indexed keywords

APPROXIMATION THEORY; GAS DYNAMICS; INTEGRAL EQUATIONS; INTEGRATION; INTERPOLATION; LAGRANGE MULTIPLIERS; MAGNETOHYDRODYNAMICS; MATRIX ALGEBRA; POLYNOMIALS; SUBSONIC FLOW; SUPERSONIC FLOW; TRANSONIC FLOW;

ADAPTIVE QUADRATURE TECHNIQUE; CONSERVATION LAWS; ENTROPY SYMMETRIZATION; EULER EQUATIONS; FLUCTUATION SPLITTING; FLUX DIVERGENCE; RESIDUAL DISTRIBUTION SCHEMES;

LINEARIZATION;

EID: 0038070198     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S106482750138592X     Document Type: Article

References (32)

1. R. Abgrall, Toward the ultimate conservative scheme: Following the quest, J. Comput. Phys., 167 (2001), pp. 277-315.
2. R. Abgrall and K. Mer, Un théorème de type Lax-Wendroff pour les schémas distributifs, Technical report 98010, Mathématiques Appliquées de Bordeaux, Université Bordeaux, Talence, France, 1998.
3. O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, UK, 1996.
4. T. J. Barth, Simplified discontinuous Galerkin methods for systems of conservation laws with convex extension, in Proceedings of the 1st International Conference on Discontinuous Galerkin Methods, Lecture Notes in Comput. Sci. Engrg. II, B. Cockburn, G. E. Karniadakis, and C.-H. Shu, eds., Springer, Berlin, 2001, pp. 63-75.
5. T. J. Barth, Some Working Notes on Energy Analysis of the Matrix N-scheme, NASA Ames Research Center, Moffett Field, CA, 1996.
6. T. J. Barth, Numerical methods for gasdynamic systems on unstructured meshes in An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, Lecture Notes in Comput. Sci. Engrg. 5, D. Kröner, M. Ohlberger, and C. Rohde, eds., Springer-Verlag, Heidelberg, 1998, pp. 195-285.
7. P. I. Crumpton, J. A. MacKenzie, and K. W. Morton, Cell vertex algorithms for the compressible Navier-Stokes equations, J. Comput. Phys., 109 (1993), pp. 1-15.
8. H. Deconinck, R. Struijs, and P. L. Roe, Compact Advection Schemes on Unstructured Grids, von Karman Institute Lecture Series Monograph 1993-04. von Karman Institute for Fluid Dynamics, Belgium, 1993.
9. A. Dervieux, B. Van Leer, J. Périaux, and A. Rizzi, Eds., Numerical Simulation of Compressible Euler Flows, Notes on Numerical Fluid Mechanics 26, Friedr. Vieweg & Sohn, Braunschweig, 1989.
10. S. K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47, (1959), pp. 271-306.
11. S. K. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk. SSSR, 139 (1961), pp. 521-523.
12. G. Golub, Private communication, Stanford University, Stanford, CA, March 11, 1996.
13. A. Harten, P. D. Lax, and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), pp. 35-61.
14. T. Y. Hou and P. G. Le Floch, Why nonconservative schemes converge to wrong solutions: Error analysis, Math. Comp., 62 (1994), pp. 497-530.
15. T. J. R. Hughes, L. P. Franca, and M. Mallet, A new finite element formulation for CFD: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics, Comput. Methods Appl. Mech. Engrg., 54 (1986), pp. 223-234.
16. T. J. R. Hughes and M. Mallet, A new finite element formulation for CFD: III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg., 58 (1986), pp. 305-328.
17. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, UK, 1987.
18. C. Johnson and A. Szepessy, Convergence of the shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp., 54 (1990), pp. 107-129.
19. D. Kröner, Numerical Schemes for Conservation Laws, John Wiley and B. G. Teubner, New York, 1997.
20. D. Kröner, M. Rokyta, and M. Wierse, A Lax-Wendroff type theorem for upwind finite volume schemes in 2-d. East-West J. Numer. Math., 4 (1996), pp. 279-292.
21. P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), pp. 217-237.
22. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, 1973.
23. M. S. Mock, Systems of conservation laws of mixed type, J. Differential Equations, 37 (1980), pp. 70-88.
24. K. W. Morton and E. Süli, Finite volume methods and their analysis, IMA J. Numer. Anal., 11 (1991), pp. 241-260.
25. R.-H. Ni, A multiple grid scheme for solving the Euler equations, AIAA J., 20 (1981), pp. 1565-1571.
26. P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43 (1981), pp. 357-372.
27. P. L. Roe, Linear Advection Schemes on Triangular Meshes, Technical Report CoA 8720, Cranfield Institute of Bioscience and Technology, Cranfield University, Bedfordshire, UK, 1987.
28. P. L. Roe, "Optimum" Upwind Advection on a Triangular Mesh, Technical report, ICASE, NASA Langley Research Center, 1990.
29. G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.
30. R. Struijs, H. Deconinck, and P. L. Roe, A multidimensional generalization of Roe's flux difference splitter for the euler equations, Comput. Fluids, 22 (1993), pp. 215-222.
31. E. van der Weide and H. Deconinck, Positive matrix distribution schemes for hyperbolic systems, in Computational Fluid Dynamics '96, Wiley, New York 1996, pp. 747-753.
32. B. van Leer, Towards the ultimate conservative difference schemes V. A second-order sequel to Godunov's method, J. Comput. Phys., 32 (1979), pp. 101-136.