FORCE schemes on unstructured meshes I: Conservative hyperbolic systems

Journal of Computational Physics

Volumn 228, Issue 9, 2009, Pages 3368-3389

  Toro, Eleuterio F ^a   Hidalgo, Arturo ^b   Dumbser, Michael ^a  

^a UNIVERSITY OF TRENTO   (Italy)

^b UNIVERSIDAD POLITÉCNICA DE MADRID   (Spain)

Author keywords

Averaging operator; Conservative hyperbolic systems; Conservative schemes; Finite elements; Finite volumes; FORCE flux; Numerical fluxes; Riemann problem; Unstructured meshes

Indexed keywords

FINITE VOLUME METHOD; MAGNETOHYDRODYNAMICS; MESH GENERATION;

AVERAGING OPERATORS; CONSERVATIVE HYPERBOLIC SYSTEM; CONSERVATIVE SCHEMES; FINITE ELEMENT; FINITE-VOLUME; FORCE FLUX; HYPERBOLIC SYSTEM; NUMERICAL FLUX; RIEMANN PROBLEM; UNSTRUCTURED MESHES;

GALERKIN METHODS;

EID: 62049084833     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2009.01.025     Document Type: Article

References (45)

1. Abgrall R. On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. Journal of Computational Physics 144 (1994) 45-58
2. Arminjon P., and St-Cyr A. Nessyahu-Tadmor-type central finite volume methods without predictor for 3D Cartesian and unstructured tetrahedral grids. Applied Numerical Mathematics 46 (2003) 135-155
3. Balsara D. Total variation diminishing scheme for relativistic magnetohydrodynamics. The Astrophysical Journal Supplement Series 132 (2001) 83-101
4. Balsara D., and Shu C.W. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. Journal of Computational Physics 160 (2000) 405-452
5. Bianco F., Puppo G., and Russo G. High order central schemes for hyperbolic systems of conservation laws. SIAM Journal on Scientific Computing 21 (1999) 294-322
6. Chen G.Q., and Toro E.F. Centred difference schemes for nonlinear hyperbolic equations. Journal of Hyperbolic Differential Equations 1 (2004) 531-566
7. Cockburn B., Hou S., and Shu C.W. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Mathematics of Computation 54 (1990) 545-581
8. Cockburn B., Karniadakis G.E., and Shu C.W. Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering (2000), Springer
9. Cockburn B., and Shu C.W. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. Journal of Computational Physics 141 (1998) 199-224
10. Dedner A., Kemm F., Kröner D., Munz C.-D., Schnitzer T., and Wesenberg M. Hyperbolic divergence cleaning for the MHD equations. Journal of Computational Physics 175 (2002) 645-673
11. Dumbser M., Balsara D., Toro E.F., and Munz C.D. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. Journal of Computational Physics 227 (2008) 8209-8253
12. Dumbser M., and Käser M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. Journal of Computational Physics 221 (2007) 693-723
13. Dumbser M., Käser M., A Titarev V., and Toro E.F. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. Journal of Computational Physics 226 (2007) 204-243
14. Dumbser M., Käser M., and Toro E.F. An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes V: Local time stepping and p-adaptivity. Geophysical Journal International 171 (2007) 695-717
15. Giacomazzo B., and Rezzolla L. The exact solution of the Riemann problem in relativistic magnetohydrodynamics. Journal of Fluid Mechanics 562 (2006) 223-259
16. Glimm J. Solution in the large for nonlinear hyperbolic systems of equations. Communications on Pure Applied Mathmatics 18 (1965) 697-715
17. Godlewski E., and Raviart P.A. Numerical Approximation of Hyperbolic Systems of Conservation Laws (1996), Springer
18. Godunov S.K., Zabrodin A.V., and Prokopov G.P. A difference scheme for two-dimensional unsteady aerodynamics. Journal of Computational Mathematics and Mathematical Physics USSR 2 6 (1961) 1020-1050
19. Godunov S.K. Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics. Mathematics of the USSR-Sbornik 47 (1959) 271-306
20. Gottlieb S., and Shu C.W. Total variation diminishing Runge-Kutta schemes. Mathematics of Computation 67 (1998) 73-85
21. Haasdonk B., Kröner D., and Rohde C. Convergence of a staggered-Lax-Friedrichs-scheme on unstructured grids in 2d. Numerische Mathematik 88 (2001) 459-484
22. Harten A. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics 49 (1983) 357-393
23. Harten A., Engquist B., Osher S., and Chakravarthy S.R. Uniformly high order accurate essentially non-oscillatory schemes III. Journal of Computational Physics 71 (1987) 231-303
24. Honkkila V., and Janhunen P. HLLC solver for ideal relativistic MHD. Journal of Computational Physics 223 (2007) 643-656
25. Hou T.Y., and LeFloch P.G. Why nonconservative schemes converge to wrong solutions: error analysis. Mathematics of Computation 62 (1994) 497-530
26. Hu C., and Shu C.W. Weighted essentially non-oscillatory schemes on triangular meshes. Journal of Computational Physics 150 (1999) 97-127
27. Jiang G.-S., and Shu C.W. Efficient implementation of weighted ENO schemes. Journal of Computational Physics (1996) 202-228
28. Jiang G.S., and Tadmor E. Non-oscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM Journal on Scientific Computing 19 (1998) 1892-1917
29. Lax P.D. Weak solutions of nonlinear hyperbolic equations and their numerical approximation. Communications on Pure Applied Mathmatics 7 (1954) 159-193
30. Lax P.D., and Wendroff B. Systems of conservation laws. Communications in Pure and Applied Mathematics 13 (1960) 217-237
31. LeVeque R.J. Finite Volume Methods for Hyperbolic Problems (2002), Cambridge University Press
32. Levy D., Puppo G., and Russo G. Central WENO schemes for hyperbolic systems of conservation laws. Mathematical Models and Numerical Analysis 33 (1999) 547-571
33. Levy D., Puppo G., and Russo G. A fourth order central WENO scheme for multidimensional hyperbolic systems of conservation laws. SIAM Journal on Scientific Computing 24 (2002) 480-506
34. Moschetta J.M., and Gressier J. A cure for the sonic point glitch. International Journal of Computational Fluids Dynamics 13 (2000) 143-159
35. Nessyahu H., and Tadmor E. Non-oscillatory central differencing for hyperbolic conservation laws. Journal of Computational Physics 87 (1990) 408-463
36. Rezzolla L., and Zanotti O. An improved exact riemann solver for relativistic hydrodynamics. Journal of Fluid Mechanics 449 (2001) 395-411
37. Rusanov V.V. Calculation of interaction of non-steady shock waves with obstacles. Journal of Computational Mathematical Physics USSR 1 (1961) 267-279
38. Sebastian K., and Shu C.W. Multidomain WENO finite difference method with interpolation at subdomain interfaces. Journal of Scientific Computing 19 (2003) 405-438
39. Shi J., Hu C., and Shu C.W. A technique of treating negative weights in WENO schemes. Journal of Computational Physics 175 (2002) 108-127
40. Titarev V.A., and Toro E.F. ADER schemes for three-dimensional nonlinear hyperbolic systems. Journal of Computational Physics 204 (2005) 715-736
41. E.F. Toro, On Glimm-related Schemes for Conservation Laws. Technical Report MMU-9602, Department of Mathematics and Physics, Manchester Metropolitan University, UK, 1996.
42. Toro E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics. second ed. (1999), Springer
43. Toro E.F., and Billet S.J. Centered TVD schemes for hyperbolic conservation laws. IMA Journal of Numerical Analysis 20 (2000) 44-79
44. Woodward P., and Colella P. The numerical simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics 54 (1984) 115-173
45. Del Zanna L., Bucciantini N., and Londrillo P. An efficient shock-capturing central-type scheme for multidimensional relativistic flows II. Magnetohydrodynamics. Astronomy and Astrophysics 400 (2003) 397-413