Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems

Authors: D. Aregba-Driollet, R. Natalini and S. Tang
Journal: Math. Comp. 73 (2004), 63-94
MSC (2000): Primary 65M06; Secondary 76M20, 76RXX, 82C40
Published electronically: August 26, 2003
MathSciNet review: 2034111
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We design numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection. These schemes are based on discrete BGK models where both characteristic velocities and the source-term depend singularly on the relaxation parameter. General stability conditions are derived, and convergence is proved to the entropy solutions for scalar equations.

References [Enhancements On Off] (What's this?)

  • 1. D. Aregba-Driollet, R. Natalini,
    Convergence of relaxation schemes for conservation laws,
    Appl. Anal. 61 (1996), 163-193.
  • 2. D. Aregba-Driollet, R. Natalini,
    Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws, SIAM J. Numer. Anal. 37 (2000), no 6, pp. 1973-2004. MR 2001f:65090
  • 3. F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models, Numer. Math. 94 (2003), 623-672.
  • 4. F. Bouchut, F. Golse, and M. Pulvirenti,
    Kinetic equations and asymptotic theory,
    Series in Appl. Math., Gauthiers-Villars, 2000.
  • 5. F. Bouchut, F. Guarguaglini, and R. Natalini, Diffusive BGK Approximations for Nonlinear Multidimensional Parabolic Equations,
    Indiana Univ. Math. J., 49:723-749, 2000. MR 2001k:35162
  • 6. J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal. 147 (1999) no 4, 269-361. MR 2000m:35132
  • 7. M. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), 1-21. MR 81b:65079
  • 8. P. D'Ancona and S. Spagnolo.
    The Cauchy problem for weakly parabolic systems.
    Math. Ann., 309:307-330, 1997. MR 98g:35119
  • 9. S.D. Eidel'man.
    Parabolic Systems.
    North-Holland Publishing Company, 1969. MR 40:6023
  • 10. Magne S. Espedal and K. Hvistendahl Karlsen, Numerical Solution of Reservoir Flow Models Based on Large Time Step Operator Splitting Algorithms, A. Fasano and H. van Duijn, editors, ``Filtration in Porous Media and Industrial Applications", Lecture Notes in Mathematics, Springer, no. 1734, pp. 3-77. MR 2002a:76115
  • 11. S. Evje and K.H. Karlsen, Monotone difference approximation of BV solutions to degenerate convection-diffusion equations, SIAM J. Numer. Anal. 37 (2000), no 6, pp. 1838-1860. MR 2001g:65110
  • 12. S. Gottlieb, C.W. Shu, and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001), 89-112 (electronic). MR 2002f:65132
  • 13. S. Gottlieb, C.W. Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp. 67 (1998), 73-85. MR 98c:65122
  • 14. B. Hanouzet, R. Natalini, Weakly coupled systems of quasilinear hyperbolic equations, Diff. Integral Eq. 9 (1996), 1279-1292. MR 97h:35138
  • 15. S. Jin, Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 122 (1995), no 1, 51-67. MR 96g:65084
  • 16. S. Jin, H.L. Liu, Diffusion limit of a hyperbolic system with relaxation, Meth. and Appl. Anal. 5 (1998), 317-334. MR 2000k:35176
  • 17. S. Jin, L. Pareschi, G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal. 35 (1998), 2405-2439. MR 99k:76100
  • 18. S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), 235-277. MR 96c:65134
  • 19. K. H. Karlsen, K.-A. Lie, J. R. Natvig, H. F. Nordhaug, and H. K. Dahle, Operator splitting methods for systems of convection-diffusion equations: nonlinear error mechanisms and correction strategies. J. of Computational Physics 173 (2001) 636-663. MR 2002h:76093
  • 20. S.N. Kruzkov, First-order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243. MR 42:2159
  • 21. A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, Journal of Computational Physics, 160 (2000) 214-282. MR 2001d:65135
  • 22. C. Lattanzio, R. Natalini, Convergence of diffusive BGK approximations for parabolic systems, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 2, 341-358. MR 2003e:35136
  • 23. P.L. Lions, G. Toscani, Diffusive limit for finite velocity Boltzmann kinetic models, Rev. Mat. Iberoamericana 13 (1997), 473-513. MR 99g:76127
  • 24. T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), 153-175. MR 88f:35092
  • 25. C. Mascia, A. Porretta, and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Ration. Mech. Anal. 163:87-124, 2002. MR 2003e:35217
  • 26. K. W. Morton, Numerical solution of convection-diffusion problems. Applied Mathematics and Mathematical Computation, 12. Chapman & Hall, London, 1996. MR 98b:65004
  • 27. R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), 795-823. MR 97c:35131
  • 28. R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, J. Diff. Eq. 148 (1998), 292-317. MR 99e:35139
  • 29. Ole{\u{\i}}\kern.15emnik, O. A. and Kruzkov, S. N., Quasi-linear parabolic second-order equations with many independent variables, Russian Math. Surveys 16 (1961), pp. 105-146. MR 25:6289
  • 30. B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property, SIAM J. Numer. Anal. 27 (1990), no 6, 1405-1421. MR 91k:65135
  • 31. C.W. Shu, S. Osher, Efficient implementation of essentially non oscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), 439-471. MR 89g:65113
  • 32. M.E. Taylor.
    Partial differential equations, III. Nonlinear equations, volume 117 of Applied Mathematical Sciences.
    Springer-Verlag, 1996. MR 98k:35001

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M06, 76M20, 76RXX, 82C40

Retrieve articles in all journals with MSC (2000): 65M06, 76M20, 76RXX, 82C40

Additional Information

D. Aregba-Driollet
Affiliation: Mathématiques Appliquées de Bordeaux, Université Bordeaux 1, 351 cours de la Libération, F-33405 Talence, France

R. Natalini
Affiliation: Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, I–00161 Roma, Italia

S. Tang
Affiliation: Department of Mechanics and Engineering Sciences, Peking University, Beijing 100871, People’s Republic of China and Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany

Received by editor(s): November 21, 2000
Received by editor(s) in revised form: January 11, 2002
Published electronically: August 26, 2003
Additional Notes: Work partially supported by European TMR projects HCL # ERB FMRX CT96 0033 and NPPDE # ERB FMRX CT98 0201, Chinese Special Funds for Major State Basic Research Project, and NSFC
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society