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Klein-Kramers Equation and Burgers/Fokker-Planck Model of Spray

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Computing Qualitatively Correct Approximations of Balance Laws

Part of the book series: SIMAI Springer Series ((SEMA SIMAI,volume 2))

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Abstract

In the preceding chapter, it has been shown that the inclusion of a Vlasov-type acceleration term inside the framework of well-balanced schemes for linear relaxation kinetic models leads to complications. There is an alternative: namely, when considering a Fokker-Planck approximation of the relaxation term, the steady-state equation can be reduced to a Sturm-Liouville eigenvalue problem. Techniques available for this class of differential equations allow for a nearly complete treatment and the spectral technique of “elementary solutions” can be set up in order to produce well- balanced schemes for which the CFL condition is affected neither by the Vlasov term, nor by the drift-diffusion term in the v variable.

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References

  1. Ball J (2003) Half-range generalized Hermite polynomials and the related Gaussian quadratures. SIAM J Numer Anal 40:2311–2317

    Article  MATH  MathSciNet  Google Scholar 

  2. Beals R (1979) On an abstract treatment of some forward-backward problems of transport and scattering. J Funct Anal 34:1–20

    Article  MATH  MathSciNet  Google Scholar 

  3. Beals R (1985) Indefinite Sturm-Liouville problems and half-range completeness. J Different Equ 56:391–407

    Article  MATH  MathSciNet  Google Scholar 

  4. Beals R, Protopopescu V (1983) Half-range completeness for the Fokker-Planck equation. J Stat Phys 32:565–584

    Article  MATH  MathSciNet  Google Scholar 

  5. Blanchard JL, Newman EH (1989) Numerical Evaluation of Parabolic Cylinder Functions. IEEE Trans on antennas and propagation 37:519–523

    Article  Google Scholar 

  6. Boyd JP (1998) Global Approximations to the Principal Real-Valued Branch of the Lambert W-function. Applied Math Lett 11:27–31

    Article  MATH  MathSciNet  Google Scholar 

  7. Boyd JP (2001) Chebyshev and Fourier Spectral Methods. Dover Publications, Mineola

    MATH  Google Scholar 

  8. Brenier Y (1990) Une méthode particulaire pour les équations non-linéaires de diffusion convection en dimension un. J Comput Applied Math 31:35–56

    Article  MATH  MathSciNet  Google Scholar 

  9. Bringuier E (2002) Fokker-Planck transport simulation tool for semiconductor devices. Phil Magazine B 82:1113–1128

    Google Scholar 

  10. Buet C, Dellacherie S (2010) On the Chang and Cooper scheme applied to a linear Fokker-Planck equation. Comm in Math Sci 8:1079–1090

    Article  MATH  MathSciNet  Google Scholar 

  11. Bunck BF (2009) A fast algorithm for evaluation of normalized Hermite functions. BIT Numer Math 49:281–295

    Article  MATH  MathSciNet  Google Scholar 

  12. Burschka MA, Titulaer UM (1981) The Kinetic Boundary Layer for the Fokker-Planck Equation with Absorbing Boundary. J Stat Phys 25:569–582

    Article  MathSciNet  Google Scholar 

  13. Burschka MA, Titulaer UM (1982) The Kinetic Boundary Layer for the Fokker-Planck Equation: A Brownian Particle in a Uniform Field. Physica 112A:315–330

    Article  Google Scholar 

  14. Cercignani C, Sgarra C (1992) Half-Range Completeness for the Fokker-Planck Equation with an External Force. J Stat Phys 66:1575–1582

    Article  MATH  MathSciNet  Google Scholar 

  15. Degond P (1986) Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions. Ann scient l’École Normale Supérieure 19:519–542

    Article  MATH  MathSciNet  Google Scholar 

  16. Degond P, Mas GS (1987) Existence of solutions and diffusion approximation for a model Fokker-Planck equation. Transp Theo Stat Phys 16:589–636

    Article  MATH  MathSciNet  Google Scholar 

  17. Domelevo K, Vignal M-H (2001) Limites visqueuses pour des systemes de type Fokker-Planck- Burgers unidimensionnels. C R Acad Sci Paris Série I Math 332:863–868

    Article  MATH  MathSciNet  Google Scholar 

  18. El-Ghani N, Masmoudi N (2010) Diffusion limit of the Vlasov-Poisson-Fokker-Planck system. Comm Math Sci 8:463–479

    Article  MATH  MathSciNet  Google Scholar 

  19. Fernandez-Feria R (1987) Fernandez de la Mora J.: Solution of the Fokker-Planck equation for the shock wave problem. J Stat Phys 48:901–917

    Article  Google Scholar 

  20. Fok JCM, Guo B, Tang T (2002) Combined Hermite spectral-finite difference method for the Fokker-Planck equation. Math Comput 71:1497–1528

    Article  MATH  MathSciNet  Google Scholar 

  21. Fisch NJ, Kruskal M (1980) Separating variables in two-way diffusion equations. J Math Phys 21:740–750

    Article  MATH  MathSciNet  Google Scholar 

  22. Goličnik M (2012) On the Lambert W function and its utility in biochemical kinetics. Biochemical Engineering Journal 63:116–123

    Article  Google Scholar 

  23. Golub GH, Welsch JH (1969) Calculation of Gauss quadrature rules. Math Comp 23:221–230

    Article  MATH  MathSciNet  Google Scholar 

  24. Gosse L, Toscani G (2006) Identification of asymptotic decay to self-similarity for one-dimensional filtration equations. SIAM J Numer Anal 43:2590–2606

    Article  MATH  MathSciNet  Google Scholar 

  25. Gosse L, Toscani G (2006) Lagrangian numerical approximations to one-dimensional convolution- diffusion equations. SIAM J Sci Comput 28:1203–1227

    Article  MATH  MathSciNet  Google Scholar 

  26. Goudon T, Saad M (1998) On a Fokker-Planck equation arising in population dynamics. Rev Math Complutense 11:353–372

    MATH  MathSciNet  Google Scholar 

  27. Greiner G (1984) Spectral Properties and Asymptotic Behavior of the Linear Transport Equation. Math Zeit 185:167–177

    Article  MATH  MathSciNet  Google Scholar 

  28. Harris S, Monroe JL (1977) Eigentheory of the inhomogeneous Fokker-Planck equation. J Stat Phys 17:377–381

    Article  Google Scholar 

  29. Hsu S-B, Liu T-P (1990) Nonlinear singular Sturm-Liouville problems and an application to transonic flow through a nozzle. Comm Pure Appl Math 43:31–61

    Article  MATH  MathSciNet  Google Scholar 

  30. Illner R (1979) I. Kuer, Collision operators as generators of Markov processes and their spectra. J Stat Phys 20:303–316

    Article  MathSciNet  Google Scholar 

  31. Jeribi A, Latrach KH, Megdiche H (2005) Time asymptotic behavior of the solution to a Cauchy problem governed by a transport operator. J Integr Equ Applic 17:121–139

    Article  MATH  MathSciNet  Google Scholar 

  32. Kim AD, Tranquilli P (2008) Numerical solution of the Fokker-Planck equation with variable coefficients. J Quant Spectr Rad Transf 109:727–740

    Article  Google Scholar 

  33. Larsen EW, Levermore CD, Pomraning GC, Sanderson JG (1985) Discretization Methods for One-Dimensional Fokker-Planck Operators. J Comput Phys 61:359–381

    Article  MATH  MathSciNet  Google Scholar 

  34. Marshall TW, Watson EJ (1985) A drop of ink falls from my pen …it comes to earth. I know not when J Phys A: Math Gen 18:3531–3559

    MATH  MathSciNet  Google Scholar 

  35. Marshall TW, Watson EJ (1987) The analytic solutions of some boundary layer problems in the theory of Brownian motion. J Phys A: Math Gen 20:1345–1354

    Article  MATH  MathSciNet  Google Scholar 

  36. Marshall TW, Watson EJ (1988) On the Fokker-Planck equation with force term. J Phys A: Math Gen 21:4241–4243

    Article  MATH  MathSciNet  Google Scholar 

  37. Mayya YS, Sahni DC (1983) One-dimensional Brownian motion near an absorbing boundary: Solution to the steady state Fokker-Planck equation. J Chem Phys 79:2302–2307

    Article  Google Scholar 

  38. Meng J., Zhang Y.: Gauss-Hermite quadratures and accuracy of lattice Boltzmann models for nonequilibrium gas flows. Physical Review E 83, id.036704 (2011)

    Google Scholar 

  39. Protopopescu V (1987) On the Fokker-Planck equation with force term. J Phys A: Math Gen 20:L1239–L1244

    Article  MATH  MathSciNet  Google Scholar 

  40. Pagani CD (1970) Studio di alcune questioni concernenti l’equazione generalizzata di Fokker- Planck. Boll Un Mat Ital 3(4):961–986

    MATH  MathSciNet  Google Scholar 

  41. Risken H (1989) The Fokker-Planck Equation: Methods of Solutions and Applications, 2nd edn. Springer Series in Synergetics, Springer, Berlin Heidelberg New York

    Book  MATH  Google Scholar 

  42. O’Rourke P.J.: Collective Drop Effects on. Vaporizing Liquid Sprays. PhD thesis. Princeton University, Princeton, NJ (1981)

    Google Scholar 

  43. Rotenberg M (1983) Transport theory for growing cell populations. J Theor Biol 103:181–199

    Article  MathSciNet  Google Scholar 

  44. Schulten Z, Gordon RG, Anderson DGM (1981) A numerical algorithm for the evaluation of Weber parabolic cylinder functions U(a;x), V(a;x), and W(a,±x). J Comput Phys 42:213–237

    Article  MATH  MathSciNet  Google Scholar 

  45. Selinger JV, Titulaer UM (1984) The Kinetic Boundary Layer for the Klein-Kramers Equation; A New Numerical Approach. J Stat Phys 36:293–318

    Article  MATH  MathSciNet  Google Scholar 

  46. Shen J, Tang T, Wang L-L (2011) Spectral Methods: Algorithms. Analysis and Applications, Springer-Verlag, Berlin Heidelberg

    Book  MATH  Google Scholar 

  47. Shizgal B (1981) A Gaussian Quadrature Procedure for Use in the Solution of the Boltzmann Equation and Related Problems. J Comput Phys 41:309–328

    Article  MATH  MathSciNet  Google Scholar 

  48. Tang T (1993) The Hermite spectral method for Gaussian-type functions. SIAM J Scient Comput 14:594–606

    Article  MATH  MathSciNet  Google Scholar 

  49. Tang T, McKee S, Reeks MW (1992) A spectral method for the numerical solutions of a kinetic equation describing the dispersion of small particles in a turbulent flow. J Comput Phys 103:222–230

    Article  MATH  Google Scholar 

  50. Van der Mee C (2008) Exponentially dichotomous operators and applications. Birkhäuser Verlag AG, Basel Boston Berlin

    MATH  Google Scholar 

  51. Van der Mee C, Zweifel P (1987) A Fokker-Planck equation for growing cell populations. J Math Biol 25:61–72

    Article  MATH  MathSciNet  Google Scholar 

  52. Whittaker ET (1902) On the Functions associated with the Parabolic Cylinder in Harmonic Analysis. Proc London Math Soc 35:417–427

    Article  MATH  MathSciNet  Google Scholar 

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  1. Istituto per le Applicazioni del Calcolo “Mauro Picone”, CNR, Rome, Italy

    Laurent Gosse

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Gosse, L. (2013). Klein-Kramers Equation and Burgers/Fokker-Planck Model of Spray. In: Computing Qualitatively Correct Approximations of Balance Laws. SIMAI Springer Series, vol 2. Springer, Milano. https://doi.org/10.1007/978-88-470-2892-0_12

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