Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Combined Hermite spectral-finite difference method for the Fokker-Planck equation


Authors: Johnson C. M. Fok, Benyu Guo and Tao Tang
Journal: Math. Comp. 71 (2002), 1497-1528
MSC (2000): Primary 65M12, 65M70; Secondary 82C31
DOI: https://doi.org/10.1090/S0025-5718-01-01365-5
Published electronically: December 21, 2001
MathSciNet review: 1933042
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The convergence of a class of combined spectral-finite difference methods using Hermite basis, applied to the Fokker-Planck equation, is studied. It is shown that the Hermite based spectral methods are convergent with spectral accuracy in weighted Sobolev space. Numerical results indicating the spectral convergence rate are presented. A velocity scaling factor is used in the Hermite basis and is shown to improve the accuracy and effectiveness of the Hermite spectral approximation, with no increase in workload. Some basic analysis for the selection of the scaling factors is also presented.


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

  • 1. M. ABRAMOWITZ AND I. R. STEGUN, Handbook of Mathematical Functions, Dover, 1972. MR 94b:00012
  • 2. R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975. MR 56:9247
  • 3. C. BERNARDI, AND Y. MADAY, Spectral methods, in Handbook of Numerical Analysis, 209-486, ed. by Ciarlet, P.G. and Lions, J.L., Elsevier, Amsterdam, 1997. CMP 98:01
  • 4. J. P. BOYD, Chebyshev and Fourier Spectral Methods, (Springer-Verlag Berlin, Heidelberg, 1989).
  • 5. J. P. BOYD, Asymptotic coefficients of Hermite function series, J. Comput. Phys., 54 (1984), p. 382. MR 86c:41012
  • 6. J. P. BOYD, Orthogonal rational functions on a semi-infinite interval, J. Comput. Phys., 70 (1987), 63-88. MR 88d:65034
  • 7. H. C. BRINKMAN, Brownian motion in a field of force and the diffusion theory of chemical reactions. Physica, 22 (1956), 29-34.
  • 8. C. BLOMBERG, The Brownian motion theory of chemical transition rates, Physica, 86A (1977), 49-66.
  • 9. M. A. BURSCHKA AND U. M. TITULAER, The kinetic boundary layer for the Fokker-Planck equation: a Brownian particle in an unbounded field, Physica, 112A, (1982), 315-330.
  • 10. M. A. BURSCHKA AND U. M. TITULAER, The kinetic boundary layer for the Fokker-Planck equation: selectively absorbing boundaries, J. Stat. Phys. 26, (1981) 59-71. MR 83b:82108
  • 11. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988. MR 89m:76004
  • 12. B. CARTLING, Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential., J. Chem. Phys., 87 (1987), 2638-2648.
  • 13. S. CHANDRASEKHAR, Stochastic problems in physics and astronomy, Rev. Mod. Phys., 15 (1943), 1-89.
  • 14. T. CHEN, A theoretical and numerical study for the Fokker-Planck equation, MSc Thesis, Dept of Math., Simon Fraser University, B. C., Canada, 1992.
  • 15. R. J. DIPERNA AND P. L. LIONS, On the Fokker-Planck-Boltzmann equation, Commun. Math. Phys., 120 (1988), 1-23. MR 90b:35203
  • 16. D. FUNARO, Polynomial Approximation of Differential Equations, Springer-Verlag, Berlin, 1992. MR 94c:65078
  • 17. D. FUNARO AND O. KAVIAN, Approximation of some diffusion evolution equations in unbounded domain by Hermite functions, Math. Comp., 57 (1991), 597-619. MR 92k:35156
  • 18. R. R. J. GAGNE AND M. M. SHOUCRI, A splitting scheme for the numerical solution of a one-dimensional Vlasov equation, J. Comput. Phys., 24 (1977), 445.
  • 19. D. GOTTLIEB AND S. ORSZAG, Numerical Analysis of Spectral Methods, Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 26, SIAM , Philadelphia, 1997. MR 58:24983
  • 20. C. E. GROSCH AND S. A. ORSZAG, Numerical solution of problems in unbounded regions: Coordinates transfroms, J. Comput. Phys., 25 (1977), 273-295. MR 58:8372
  • 21. B. GUO, Finite Difference Methods for Partial Differential Equations, Science Press, Beijing, 1988.
  • 22. B. GUO, Spectral Methods and Their Applications, World Scientific, River Edge, NJ, 1998. MR 2000b:65194
  • 23. B. GUO, Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comp., 68, (1999), 1067-1078. MR 99i:65111
  • 24. B. Y. GUO AND J. SHEN, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math., 86, 2000, 635-654. MR 2001h:65152
  • 25. D. W. HEERMAN, Computer Simulation Methods in Theoretical Physics, Springer Verlag, Berlin, 1986.
  • 26. J. P. HOLLOWAY, Spectral velocity discretizations for the Vlasov-Maxwell equations, Trans. Theory and Stat. Phys., 25 (1996), 1-32. MR 96k:82071
  • 27. G. JOYCE, G. KNORR AND H. K. MEIER, Numerical integration methods of the Vlasov equation. J. Comput. Phys., 8 (1971), 53-63.
  • 28. H. A. KRAMERS, Brownian motion in a field force and the diffusion of chemical reactions, Physica, 7 (1940), 284-304. MR 2:140d
  • 29. A. L. LEVIN AND D. S. LUBINSKY, Christoffel functions, orthogonal polynomials, and Nevais conjecture for Freud weights, Constr. Approx., 8 (1992), 461-535. MR 94f:42030
  • 30. D. S. LUBINSKY AND F. MORITZ, The weighted $L_p$-norms of orthogonal polynomials for Freud weights, J. Approx. Theory, 77 (1994), 42-50.
  • 31. Y. MADAY, B. PERNAUD-THOMAS, AND H. VANDEVEN, Une réhabilitation des méthodes spèctrales de type Laguerre, Rech. Aérospat., 6, 1985, 353-375. MR 88b:65135
  • 32. C. MAVRIPLIS, Laguerre polynomials for infinite-domain spectral elements, J. Comp. Phys., 80 (1989), 480-488. MR 90f:65228
  • 33. P. MOORE AND J. FLAHERTY, Adaptive local overlapping grid methods for parabolic systems in two space dimensions, J. Comp. Phys., 98 (1992), 54-63.
  • 34. B. PERTHAME, Higher moments for kinetic equations, The Vlasov-Poisson and Fokkel-Planck cases, Math. Meth. App. Sci., 13 (1990), 441-452. MR 91j:82044
  • 35. H. RISKEN, The Fokker-Planck equation: Methods of solution and applications, 2nd ed., Springer-Verlag, Berlin, 1989. MR 90a:82002
  • 36. J. W. SCHUMER AND J. P. HOLLOWAY, Vlasov simulations using velocity-scaled Hermite representations. J. Comp. Phys, 144, (1998), 626-661.
  • 37. J. SHEN, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), 1113-1133. MR 2001g:65165
  • 38. G. SZEG¨O, Orthogonal Polynomials, Amer. Math. Soc., New York, 1959. MR 21:5029
  • 39. T. TANG, The Hermite spectral method for Gaussian type funcitons, SIAM J. Sci. Comp., 14, (1993), 594-606. MR 93m:65026
  • 40. T. TANG, S. MCKEE, AND M. W. REEKS, A spectral method for the numerical solutions of a kinetic equation describing the dspersion of small particles in a turbulent flow, J. Comp. Phys, 103, (1991), 222-230.
  • 41. R. TSEMAN, Analysis Numérique, Presses Universitaires de France, Paris, 1970.
  • 42. V. THOM´EE, Difference methods for two-dimensional mixed problems for hyperbolic first order systems, Arch. Rat. Mech. Anal., 8 (1961), 68-88. MR 23:B2591
  • 43. A. F. TIMAN, Theory of Approximation of Functions of a Real Variable, Pergamon Press, Oxford, 1963. MR 33:465
  • 44. K. VOIGTLAENDER AND H. RISEN, Eigenvalues of the Fokker-Planck and BGK operators for a double-well potential, Chem. Phys. Lett., 105 (1984), 506-510.
  • 45. K. VOIGTLAENDER AND H. RISEN, Solutions of the Fokker-Planker equation for a double-well potential in terms of matrix continued fractions, J. Stat. Phys., 40 (1985), 397-429.
  • 46. J. A. C. WEIDEMAN, The eigenvalues of Hermite and rational spectral differentiation matrices, Numer. Math., 61 (1992), 409-431. MR 92k:65071

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M12, 65M70, 82C31

Retrieve articles in all journals with MSC (2000): 65M12, 65M70, 82C31


Additional Information

Johnson C. M. Fok
Affiliation: Department of Mathematics, The Hong Kong Baptist University, Kowloon Tong, Hong Kong
Email: cmfok@math.hkbu.edu.hk

Benyu Guo
Affiliation: Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China
Email: byguo@guomai.sh.cn

Tao Tang
Affiliation: Department of Mathematics, The Hong Kong Baptist University, Kowloon Tong, Hong Kong.
Email: ttang@math.hkbu.edu.hk

DOI: https://doi.org/10.1090/S0025-5718-01-01365-5
Keywords: Fokker-Planck equation, unbounded domain, Hermite spectral method, finite-difference method, error analysis
Received by editor(s): December 13, 1999
Received by editor(s) in revised form: October 30, 2000
Published electronically: December 21, 2001
Additional Notes: This research was partially supported by FRG Grants of Hong Kong Baptist University and RGC Grants of Hong Kong Research Grants Council.
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society