Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. (English) Zbl 1031.65093
Numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection are described. 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.
Reviewer: Leonid B.Chubarov (Novosibirsk)
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 35K55 | Nonlinear parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 76R50 | Diffusion |
Keywords:
nonlinear degenerate parabolic systems; discrete BGK models; stability; convergence; entropy solutionsReferences:
| [1] | D. Aregba-Driollet, R. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal. 61 (1996), 163-193. · Zbl 0872.65086 |
| [2] | Denise Aregba-Driollet and Roberto Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws, SIAM J. Numer. Anal. 37 (2000), no. 6, 1973 – 2004. · Zbl 0964.65096 · doi:10.1137/S0036142998343075 |
| [3] | F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models, Numer. Math. 94 (2003), 623-672. · Zbl 1029.65092 |
| [4] | F. Bouchut, F. Golse, and M. Pulvirenti, Kinetic equations and asymptotic theory, Series in Appl. Math., Gauthiers-Villars, 2000. · Zbl 0979.82048 |
| [5] | F. Bouchut, F. R. Guarguaglini, and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations, Indiana Univ. Math. J. 49 (2000), no. 2, 723 – 749. · Zbl 0964.35011 · doi:10.1512/iumj.2000.49.1811 |
| [6] | José Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal. 147 (1999), no. 4, 269 – 361. · Zbl 0935.35056 · doi:10.1007/s002050050152 |
| [7] | Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1 – 21. · Zbl 0423.65052 |
| [8] | Piero D’Ancona and Sergio Spagnolo, The Cauchy problem for weakly parabolic systems, Math. Ann. 309 (1997), no. 2, 307 – 330. · Zbl 0891.35050 · doi:10.1007/s002080050114 |
| [9] | S. D. Èĭdel\(^{\prime}\)man, Parabolic systems, Translated from the Russian by Scripta Technica, London, North-Holland Publishing Co., Amsterdam-London; Wolters-Noordhoff Publishing, Groningen, 1969. |
| [10] | Magne S. Espedal and Kenneth Hvistendahl Karlsen, Numerical solution of reservoir flow models based on large time step operator splitting algorithms, Filtration in porous media and industrial application (Cetraro, 1998) Lecture Notes in Math., vol. 1734, Springer, Berlin, 2000, pp. 9 – 77. · Zbl 1077.76546 · doi:10.1007/BFb0103975 |
| [11] | Steinar Evje and Kenneth Hvistendahl Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations, SIAM J. Numer. Anal. 37 (2000), no. 6, 1838 – 1860. · Zbl 0985.65100 · doi:10.1137/S0036142998336138 |
| [12] | Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001), no. 1, 89 – 112. · Zbl 0967.65098 · doi:10.1137/S003614450036757X |
| [13] | Sigal Gottlieb and Chi-Wang Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp. 67 (1998), no. 221, 73 – 85. · Zbl 0897.65058 |
| [14] | Roberto Natalini and Bernard Hanouzet, Weakly coupled systems of quasilinear hyperbolic equations, Differential Integral Equations 9 (1996), no. 6, 1279 – 1292. · Zbl 0879.35093 |
| [15] | Shi Jin, Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 122 (1995), no. 1, 51 – 67. · Zbl 0840.65098 · doi:10.1006/jcph.1995.1196 |
| [16] | Shi Jin and Hailiang Liu, Diffusion limit of a hyperbolic system with relaxation, Methods Appl. Anal. 5 (1998), no. 3, 317 – 334. · Zbl 0922.35086 · doi:10.4310/MAA.1998.v5.n3.a6 |
| [17] | Shi Jin, Lorenzo Pareschi, and Giuseppe Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal. 35 (1998), no. 6, 2405 – 2439. · Zbl 0938.35097 · doi:10.1137/S0036142997315962 |
| [18] | Shi Jin and Zhou Ping Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), no. 3, 235 – 276. · Zbl 0826.65078 · doi:10.1002/cpa.3160480303 |
| [19] | K. Hvistendahl 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. Comput. Phys. 173 (2001), no. 2, 636 – 663. · Zbl 1051.76046 · doi:10.1006/jcph.2001.6901 |
| [20] | S. N. Kružkov, First order quasilinear equations with several independent variables., Mat. Sb. (N.S.) 81 (123) (1970), 228 – 255 (Russian). |
| [21] | Alexander Kurganov and Eitan Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys. 160 (2000), no. 1, 241 – 282. · Zbl 0987.65085 · doi:10.1006/jcph.2000.6459 |
| [22] | Corrado Lattanzio and Roberto Natalini, Convergence of diffusive BGK approximations for nonlinear strongly parabolic systems, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 2, 341 – 358. · Zbl 1043.35075 · doi:10.1017/S0308210500001669 |
| [23] | Pierre Louis Lions and Giuseppe Toscani, Diffusive limit for finite velocity Boltzmann kinetic models, Rev. Mat. Iberoamericana 13 (1997), no. 3, 473 – 513. · Zbl 0896.35109 · doi:10.4171/RMI/228 |
| [24] | Tai-Ping Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), no. 1, 153 – 175. · Zbl 0633.35049 |
| [25] | Corrado Mascia, Alessio Porretta, and Andrea Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Ration. Mech. Anal. 163 (2002), no. 2, 87 – 124. · Zbl 1027.35081 · doi:10.1007/s002050200184 |
| [26] | K. W. Morton, Numerical solution of convection-diffusion problems, Applied Mathematics and Mathematical Computation, vol. 12, Chapman & Hall, London, 1996. · Zbl 0861.65070 |
| [27] | Roberto Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), no. 8, 795 – 823. , https://doi.org/10.1002/(SICI)1097-0312(199608)49:83.0.CO;2-3 · Zbl 0872.35064 |
| [28] | Roberto Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, J. Differential Equations 148 (1998), no. 2, 292 – 317. · Zbl 0911.35073 · doi:10.1006/jdeq.1998.3460 |
| [29] | Oleĭ\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. · Zbl 0112.32604 |
| [30] | B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property, SIAM J. Numer. Anal. 27 (1990), no. 6, 1405 – 1421 (English, with French summary). · Zbl 0714.76078 · doi:10.1137/0727081 |
| [31] | Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439 – 471. · Zbl 0653.65072 · doi:10.1016/0021-9991(88)90177-5 |
| [32] | Michael E. Taylor, Partial differential equations. III, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Nonlinear equations; Corrected reprint of the 1996 original. |
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