TVD fluxes for the high-order ADER schemes

Journal of Scientific Computing

Volumn 24, Issue 3, 2005, Pages 285-309

  Toro, E F ^a   Titarev, V A ^a  

^a UNIVERSITY OF TRENTO   (Italy)

Author keywords

ADER; Essentially non oscillatory; Flux expansion; Generalised Riemann problem; High order schemes; TVD flux; WAF flux

Indexed keywords

GAIN MEASUREMENT;

ADER; ESSENTIALLY NON-OSCILLATORY; FLUX EXPANSION; HIGH-ORDER SCHEMES; TVD FLUX; WAF FLUX;

NATURAL SCIENCES COMPUTING;

EID: 27244448103     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-004-4790-8     Document Type: Article

References (30)

1. Balsara D. S., and Shu, C. W. (2000). Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys., 160, 405-452.
2. Ben-Artzi M., and Falcovitz, J. (1984). A second order Godunov-type scheme for compressible fluid dynamics. J. Comput. Phys. 55, 1-32.
3. Billett, S. J., and Toro, E. F. (1997). WAF-type schemes for multidimensional hyperbolic conservation laws. J. Comp. Phys. 130, 1-24.
4. Colella, P. (1985). A direct eulerian MUSCL scheme for gas dynamics. SIAM J. Sci. Stat. Comput. 6, 104-117.
5. Godunov, S. K. (1959). Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 271-306.
6. Harten, A., Engquist, B., Osher, S., and Chakravarthy, S. R. (1987). Uniformly high order accuracy essentially non-oscillatory schemes III. J. Comput. Phys. 71, 231-303.
7. Jiang G. S., and Shu, C. W. (1995). Efficient implementation of weigthed ENO schemes. Technical Report ICASE 95-73, NASA Langley Research Center, Hampton, USA.
8. Käser, M. (2003). Adaptive Methods for the Numerical Simulation of Transport Processes. PhD thesis, Institute of Numerical Mathematics and Scientific Computing, University of Munich, Germany.
9. Käser, M. (2004). ADER Schemes for the solution of conservation laws on adaptive triangulations. Mathematical Methods and Modelling in Hydrocarbon Exploration and Production. Springer-Verlag, Berlin, (to appear).
10. Men'shov, I. S. (1990). Increasing the order of approximation of godunov's scheme using the generalized riemann problem. USSR Comput. Math. Phys. 30(5), 54-65.
11. Schwartzkopff, T., Munz, CD, and Toro, E. F. (2002). ADER: high-order approach for linear hyperbolic systems in 2D. J. Sci. Comput. 17, 231-240.
12. Schwartzkopff, T., Munz, C. D., Toro, E. F., and Millington, R. C. (2001). ADER-2D: A high-order approach for linear hyperbolic systems in 2D. In European Congress on Computational Methods in Applied Sciences and Engineering. ECCOMAS Computational Fluid Dynamics Conference 2001, Swansea, Wales, 4-7 September 2001, pp. -, 2001.
13. Shu, C. W. (1986). Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comp. 9, 1073-1084.
14. Shu, C. W. (1997). Essentially Non-oscillatory and Weighted Non-oscillatory Schemes for Hyperbolic Conservation Laws. Technical Report ICASE Report No. 97-65, NASA.
15. Suresh, A., and Huynh, T. (1997). Accurate monotonicity preserving scheme using Runge-Kutta time stepping. J. Comput. Phys., 136, 83-99.
16. Takakura, Y., and Toro, E. F. (2002). Arbitrarily accurate non-oscillatory schemes for a non-linear conservation law. J. Compu. Fluid Dyn. 11(1):7-18.
17. Titarev, V. A., and Toro, E. F. (2002). ADER: Arbitrary high order Godunov approach. J. Sci. Comput., 17, 609-618.
18. Titarev, V. A., and Toro, E. F. (2003). High-order ADER schemes for scalar advection-reaction-diffusion equations. CFD J. 12(1), 1-6.
19. Toro, E. F. (1989). A weighted average flux method for hyperbolic conservation laws. Proc. Roy. Soc. London A423, 401-418.
20. Toro, E. F. (1992). Riemann problems and the WAF method for solving two-dimensional shallow water equations. Phil. Trans. Roy. Soc. London, A338: 43-68.
21. Toro, E. F (1992). The weighted average flux method applied to the time-dependent euler equations. Phil. Trans. Roy. Soc. London, A341, 499-530.
22. Toro, E. F. (1998). Primitive, conservative and adaptive schemes for hyperbolic conservation laws. In Toro, E. F. and Clarke, J. F. (ed.), Numerical Methods for Wave Propagation. Kluwer Academic Publishers, Dordrecht, pp. 323-385.
23. Toro, E. F. (1999). Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd ed., Springer-Verlag, Berlin.
24. Toro, E. F., Millington, R. C. and Nejad, L. A. M. (1998) Primitive upwind numerical methods for hyperbolic partial differential equations. In Bruneau, C. H. (ed.), Sixteenth International Conference on Numerical Methods for Fluid Dynamics. Lecture Notes in Physics, Springer-Verlag, Berlin, pp. 421-426.
25. Toro, E. F., Millington, R. C. and Nejad, L. A. M. (2001). Towards very high-order Godunov schemes. In Toro, E. F. (ed.), Godunov Methods: Theory and Applications. Edited Review, Kluwer Academic/Plenum Publishers, Dordrecht, pp. 905-937.
26. Toro, E. F., Spruce, M., and Speares, W. (1992). Restoration of the contact surface in the HLL Riemann solver. Technical Report COA-9204, College of Aeronautics, Cranfield Institute of Technology, UK.
27. Toro, E. F., Spruce, M., and Speares, W. (1994). Restoration of the contact surface in the HLL-Riemann solver. Shock Waves., 4, 25-34.
28. Toro, E. F., and Titarev, V. A. (2002). Solution of the generalised Riemann problem for advection-reaction equations. Proc. Roy. Soc. London A, 458, 271-281.
29. Toro, E. F., and Titarev, V. A. (2003). ADER schemes for scalar hyperbolic conservation laws in three space dimensions. Technical Report NI03063-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK.
30. Woodward, P., and Colella, P. (1984). The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115-173.