An exact particle method for scalar conservation laws and its application to stiff reaction kinetics

Lecture Notes in Computational Science and Engineering

Volumn 79 LNCSE, Issue , 2011, Pages 105-124

  Farjoun, Yossi ^a   Seibold, Benjamin ^b  

^a UNIVERSIDAD CARLOS III DE MADRID   (Spain)

^b Temple University   (United States)

Author keywords

characteristic; particle; reaction kinetics; shock

Indexed keywords

BALANCE LAW; CHARACTERISTIC; CONSERVATION LAW; DETONATION WAVES; HYPERBOLIC CONSERVATION LAWS; INTERNAL DYNAMICS; LOCAL-SIMILARITY SOLUTIONS; NUMERICAL ACCURACY; NUMERICAL SOLUTION; ODE SOLVER; PARTICLE; PARTICLE METHODS; SCALAR CONSERVATION LAWS; SHOCK;

ASSOCIATION REACTIONS; PARTIAL DIFFERENTIAL EQUATIONS; PHYSICAL PROPERTIES;

REACTION KINETICS;

EID: 78651589658     PISSN: 14397358     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-16229-9_7     Document Type: Conference Paper

References (24)

1. B. Bukiet, J. Pelesko, X. L. Li, and P. L. Sachdev, A characteristic based numerical method for nonlinear wave equations, Computers Math. Applic., 31 (1996), pp. 75-79. Clawpack. Website. http://www.clawpack.org.
2. G. M. Coclite, M. Garavello, and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), pp. 1862-1886.
3. R. Courant, K. Friedrichs, and H. Lewy, Uber die partiellen Differenzen-gleichungen der mathematischen Physik, Mathematische Annalen, 100 (1928), pp. 32-74.
4. R. Courant, E. Isaacson, and M. Rees, On the solution of nonlinear hyperbolic differential equations by finite differences, Comm. Pure Appl. Math., 5 (1952), pp. 243-255.
5. J. R. Dormand and P. J. Prince, Runge-Kutta triples, Comp. Math. Appl., 12 (1986), pp. 1007-1017.
6. L. C. Evans, Partial differential equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, 1998.
7. H. Fan, S. Jin, and Z.-H. Teng, Zero reaction limit for hyperbolic conservation laws with source terms, J. Diff. Equations, 168 (2000), pp. 270-294.
8. Y. Farjoun and B. Seibold, Solving one dimensional scalar conservation laws by particle management, in Meshfree methods for Partial Differential Equations IV, M. Griebel and M. A. Schweitzer, eds., vol. 65 of Lecture Notes in Computational Science and Engineering, Springer, 2008, pp. 95-109.
9. An exactly conservative particle method for one dimensional scalar conservation laws, J. Comput. Phys., 228(2009) 5298-5315.
10. A rarefaction-tracking method for conservation laws, J. Eng. Math, 66(2010), 237-251.
11. W. Fickett and W. C. Davis, Detonation, Univ. of California Press, Berkeley, CA, 1979.
12. S. K. Godunov, A difference scheme for the numerical computation of a discontinuous solution of the hydrodynamic equations, Math. Sbornik, 47 (1959), pp. 271-306.
13. A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes. III, J. Comput. Phys., 71 (1987), pp. 231-303.
14. C. Helzel, R. J. LeVeque, and G. Warnecke, A modified fractional step method for the accurate approximation of detonation waves, SIAM J. Sci. Com-put., 22 (2000), pp. 1489-1510.
15. M. Herty and A. Klar, Modelling, simulation and optimization of traffic flow networks, SIAM J. Sci. Comp., 25 (2003), pp. 1066-1087.
16. H. Holden, L. Holden, and R. Hegh-Krohn, A numerical method for first order nonlinear scalar conservation laws in one dimension, Comput. Math. Appl., 15 (1988), pp. 595-602.
17. H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), pp. 999-1017.
18. Front Tracking for Hyperbolic Conservation Laws, Springer, 2002.
19. P. D. Lax and B. Wendroff, Systems of conservation laws, Commun. Pure Appl. Math., 13 (1960), pp. 217-237.
20. R. J. Le Veque, Finite volume methods for hyperbolic problems, Cambridge University Press, first ed., 2002.
21. M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, vol. 229 of Proc. Roy. Soc. A, Piccadilly, London, 1955, pp. 317-345.
22. X.-D. Liu, S. Osher, and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), pp. 200-212.
23. L. Shampine and M. W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput., 18 (1997), pp. 1-22.
24. B. van Leer, Towards the ultimate conservative difference scheme II. Mono-tonicity and conservation combined in a second order scheme, J. Comput. Phys., 14 (1974), pp. 361-370.