Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems

Journal of Computational Physics

Volumn 221, Issue 2, 2007, Pages 693-723

  Dumbser, Michael ^a,b   Käser, Martin ^b  

^a UNIVERSITY OF STUTTGART   (Germany)

^b UNIVERSITY OF TRENTO   (Italy)

Author keywords

ADER approach; Finite volume schemes; Linear hyperbolic systems; Unstructured meshes in two and three space dimensions; WENO reconstruction

Indexed keywords

COMPUTATIONAL METHODS;

ADER APPROACH; FINITE VOLUME SCHEMES; HIGH-ORDER; LINEAR HYPERBOLIC SYSTEMS; SPACE AND TIME; THREE SPACE DIMENSIONS; UNSTRUCTURED GRID; UNSTRUCTURED MESH IN TWO AND THREE SPACE DIMENSION; UNSTRUCTURED MESHES; WENO RECONSTRUCTION;

GALERKIN METHODS;

EID: 33846336138     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2006.06.043     Document Type: Article

References (48)

1. Abgrall R. On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. Journal of Computational Physics 144 (1994) 45-58
2. Atkins H., and Shu C.W. Quadrature-free implementation of the discontinuous Galerkin method for hyperbolic equations. AIAA Journal 36 (1998) 775-782
3. 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
4. T.J. Barth, P.O. Frederickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA paper no. 90-0013, 28th Aerospace Sciences Meeting January 1990.
5. Ben-Artzi M., and Falcovitz J. A second-order Godunov-type scheme for compressible fluid dynamics. Journal of Computational Physics 55 (1984) 1-32
6. Butcher J.C. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (1987), Wiley
7. Casper J., and Atkins H.L. A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems. Journal of Computational Physics 106 (1993) 62-76
8. Cockburn B., Karniadakis G.E., and Shu C.W. Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering (2000), Springer
9. Davies-Jones R. Comments on 'a kinematic analysis of frontogenesis associated with a non-divergent vortex'. Journal of Atmospheric Sciences 42 (1985) 2073-2075
10. Dubiner M. Spectral methods on triangles and other domains. Journal of Scientific Computing 6 (1991) 345-390
11. Dumbser M. Arbitrary High Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains (2005), Shaker Verlag, Aachen
12. M. Dumbser, M. Käser, An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes II: The three-dimensional isotropic case, Geophysical Journal International, in press.
13. Dumbser M., and Munz C.D. ADER discontinuous Galerkin schemes for aeroacoustics. Comptes Rendus Mécanique 333 (2005) 683-687
14. Dumbser M., and Munz C.D. Arbitrary high order discontinuous Galerkin schemes. In: Cordier S., Goudon T., Gutnic M., and Sonnendrucker E. (Eds). Numerical Methods for Hyperbolic and Kinetic Problems. IRMA Series in Mathematics and Theoretical Physics (2005), EMS Publishing House 295-333
15. Dumbser M., and Munz C.D. Building blocks for arbitrary high order discontinuous Galerkin schemes. Journal of Scientific Computing 27 (2006) 215-230
16. Dumbser M., Schwartzkopff T., and Munz C.D. Arbitrary high order finite volume schemes for linear wave propagation. Computational Science and High Performance Computing II. Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM) (2006), Springer 129-144
17. Friedrich O. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. Journal of Computational Physics 144 (1998) 194-212
18. 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
19. Harten A. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics 49 (1983) 357-393
20. 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
21. Hempel D. Local mesh adaption in two space dimensions. IMPACT Computational Science & Engineering 5 (1993) 309-317
22. D. Hempel, Isotropic refinement and recoarsening in 2 dimensions, Technical Report DLR IB 223-95 A 35, Deutsches Zentrum fnr Luft- und Raumfahrt (DLR), 1995.
23. Hu C., and Shu C.W. Weighted essentially non-oscillatory schemes on triangular meshes. Journal of Computational Physics 150 (1999) 97-127
24. Doswell III C.A. A kinematic analysis of frontogenesis associated with a non-divergent vortex. Journal of Atmospheric Sciences 41 (1984) 1242-1248
25. Jiang G.-S., and Shu C.W. Efficient implementation of weighted ENO schemes. Journal of Computational Physics (1996) 202-228
26. Käser M., and Dumbser M. An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes I: The two-dimensional isotropic case with external source terms. Geophysical Journal International 166 (2006) 855-877
27. Käser M., and Iske A. ADER schemes on adaptive triangular meshes for scalar conservation laws. Journal of Computational Physics 205 (2005) 486-508
28. Meister A., and Struckmeier J. Hyperbolic Partial Differential Equations (2002), Vieweg
29. Munz C.D. On the numerical dissipation of high resolution schemes for hyperbolic conservation laws. Journal of Computational Physics 77 (1988) 18-39
30. Ollivier-Gooch C., and Van Altena M. A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation. Journal of Computational Physics 181 (2002) 729-752
31. Qiu J., and Shu C.W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case. Journal of Computational Physics 193 (2003) 115-135
32. Qiu J., and Shu C.W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case. Computers and Fluids 34 (2005) 642-663
33. Qiu J., and Shu C.W. Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM Journal on Scientific Computing 26 (2005) 907-929
34. Schwartzkopff T., Dumbser M., and Munz C.D. Fast high order ADER schemes for linear hyperbolic equations. Journal of Computational Physics 197 (2004) 532-539
35. Schwartzkopff T., Munz C.D., and Toro E.F. ADER: a high order approach for linear hyperbolic systems in 2d. Journal of Scientific Computing 17 1-4 (2002) 231-240
36. 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
37. Sonar T. On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection. Computer Methods in Applied Mechanics and Engineering 140 (1997) 157-181
38. Stroud A.H. Approximate Calculation of Multiple Integrals (1971), Prentice-Hall Inc., Englewood Cliffs, NJ
39. Sweby P.K. High resolution TVD schemes using flux limiters. Lecture Notes in Applied Mathematics 22 (1985) 289-309
40. Titarev V.A., and Toro E.F. ADER: arbitrary high order Godunov approach. Journal of Scientific Computing 17 1-4 (2002) 609-618
41. Titarev V.A., and Toro E.F. ADER schemes for three-dimensional nonlinear hyperbolic systems. Journal of Computational Physics 204 (2005) 715-736
42. Toro E.F., Millington R.C., and Nejad L.A.M. Towards very high order Godunov schemes. In: Toro E.F. (Ed). Godunov Methods. Theory and Applications (2001), Kluwer/Plenum Academic Publishers 905-938
43. Toro E.F., and Titarev V. TVD fluxes of the high-order ADER schemes. Journal of Scientific Computing 24 (2005) 285-309
44. Toro E.F., and Titarev V.A. Solution of the generalized Riemann problem for advection-reaction equations. Proceedings of Royal Society of London (2002) 271-281
45. Toro E.F., and Titarev V.A. ADER schemes for scalar hyperbolic conservation laws with source terms in three space dimensions. Journal of Computational Physics 202 (2005) 196-215
46. van Leer B. Towards the ultimate conservative difference scheme II: Monotonicity and conservation combined in a second order scheme. Journal of Computational Physics 14 (1974) 361-370
47. van Leer B. Towards the ultimate conservative difference scheme V: a second order sequel to Godunov's method. Journal of Computational Physics 32 (1979) 101-136
48. Zalesak S.T. Fully multidimensional flux-corrected transport algorithms for fluids. Journal of Computational Physics 31 (1979) 335-362