Top

Journal of Scientific Computing

Published in:

01-03-2015

The Entropy Satisfying Discontinuous Galerkin Method for Fokker–Planck equations

Authors: Hailiang Liu, Hui Yu

Published in: Journal of Scientific Computing | Issue 3/2015

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In Liu and Yu (SIAM J Numer Anal 50(3):1207–1239, 2012), we developed a finite volume method for Fokker–Planck equations with an application to finitely extensible nonlinear elastic dumbbell model for polymers subject to homogeneous fluids. The method preserves positivity and satisfies the discrete entropy inequalities, but has only first order accuracy in general cases. In this paper, we overcome this problem by developing uniformly accurate, entropy satisfying discontinuous Galerkin methods for solving Fokker–Planck equations. Both semidiscrete and fully discrete methods satisfy two desired properties: mass conservation and entropy satisfying in the sense that these schemes are shown to satisfy the discrete entropy inequality. These ensure that the schemes are entropy satisfying and preserve the equilibrium solutions. It is also proved the convergence of numerical solutions to the equilibrium solution as time becomes large. At the finite time, a positive truncation is used to generate the nonnegative numerical approximation which is as accurate as the obtained numerical solution. Both one and two-dimensional numerical results are provided to demonstrate the good qualities of the schemes, as well as effects of some canonical homogeneous flows.

previous article Runge–Kutta Residual Distribution Schemes

next article A Level Set Approach Reflecting Sheet Structure with Single Auxiliary Function for Evolving Spirals on Crystal Surfaces

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Ammar, A., Mokdad, B., Chinesta, F., Keunings, R.: A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J. Non-Newton. Fluid Mech. 139(3), 153–176 (2006)CrossRefMATH

Ammar, A., Mokdad, B., Chinesta, F., Keunings, R.: A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids Part II: Transient simulation using space-time separated representations. J. Non-Newton. Fluid Mech. 144(2–3), 98–121 (2007)CrossRefMATH

Arnold, A., Carrillo, J.A., Manzini, C.: Refined long-time asymptotics for some polymeric fluid flow models. Commun. Math. Sci. 8(3), 763–782 (2010)CrossRefMATHMathSciNet

Arnold, A., Markowich, P., Toscani, G., Unterreiter, A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations. Commun. Partial Differ. Equ. 26(1–2), 43–100 (2001)CrossRefMATHMathSciNet

Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)CrossRefMATHMathSciNet

Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5):1749–1779, (2001/02)

Baker, G.A.: Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31, 45–59 (1977)CrossRefMATH

Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131(2), 267–279 (1997)CrossRefMATHMathSciNet

Baumann, C.E., Oden, J.T.: A discontinuous \(hp\) finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 175(3–4), 311–341 (1999)CrossRefMATHMathSciNet

10.

Buet, C., Dellacherie, S.: On the Chang and Cooper scheme applied to a linear Fokker–Planck equation. Commun. Math. Sci. 8(4), 1079–1090 (2010)CrossRefMATHMathSciNet

11.

Buet, C., Dellacherie, S., Sentis, R.: Numerical solution of an ionic Fokker–Planck equation with electronic temperature. SIAM J. Numer. Anal. 39(4), 1219–1253 (2001). (electronic)CrossRefMATHMathSciNet

12.

Barrett, J.W., Süli, E.: Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers. ESAIM M2AN 46, 949–978 (2012)CrossRefMATH

13.

Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38(5), 1676–1706 (2000). (electronic)CrossRefMATHMathSciNet

14.

Chang, J.S., Cooper, G.: A practical difference scheme for Fokker–Planck equations. J. Comput. Phys. 6(1), 1–16 (1970)CrossRefMATH

15.

Chauvière, C., Lozinski, A.: Simulation of complex viscoelastic flows using Fokker–Planck equation: 3D FENE model. J. Non-Newton. Fluid Mech. 122(1–3), 201–214 (2004)CrossRefMATH

16.

Chauvière, C., Lozinski, A.: Simulation of dilute polymer solutions using a Fokker–Planck equation. J. Comput. Fluids 33(5–6), 687–696 (2004)CrossRefMATH

17.

Cheng, Y., Shu, C.-W.: A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Math. Comput. 77(262), 699–730 (2007)CrossRefMathSciNet

18.

Chupin, L.: The FENE model for viscoelastic thin film flows. Methods Appl. Anal. 16(2), 217–261 (2009)MATHMathSciNet

19.

Chupin, L.: The FENE viscoelastic model and thin film flows. C. R. Math. Acad. Sci. Paris 347(17–18), 1041–1046 (2009)CrossRefMATHMathSciNet

20.

Chupin, L.: Fokker–Planck equation in bounded domain. Ann. Inst. Fourier (Grenoble) 60(1), 217–255 (2010)CrossRefMATHMathSciNet

21.

Cockburn, B., Dawson, C.: Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multi-dimensions. In: Whiteman, J.R. (ed.) Proceedings of the Conference on the Mathematics of Finite Elements and Applications, MAFELAP X, pp. 225–238. Elsevier, New York (2000)

22.

Cockburn, B., Hou, S., Shu, C.-W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54(190), 545–581 (1990)MATHMathSciNet

23.

Cockburn, B., Kanschat, G., Perugia, I., Schötzau, D.: Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 39, 264–285 (2001)CrossRefMATHMathSciNet

24.

Cockburn, B., Lin, S.Y., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)CrossRefMATHMathSciNet

25.

Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411–435 (1989)MATHMathSciNet

26.

Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)CrossRefMATHMathSciNet

27.

Du, Q., Liu, C., Yu, P.: FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul. 4(3), 709–731 (2005). (electronic)CrossRefMATHMathSciNet

28.

Fan, X.-J.: Viscosity, first normal-stress coefficient, and molecular stretching in dilute polymer solutions. J. Non-Newton. Fluid Mech. 17(2), 125–144 (1985)CrossRefMATH

29.

Gassner, G., Lörcher, F., Munz, C.-D.: A contribution to the construction of diffusion fluxes for finite volume and discontinuous galerkin schemes. J. Comput. Phys. 224(2), 1049–1063 (2007)CrossRefMATHMathSciNet

30.

Herrchen, M., Öttinger, H.C.: A detailed comparison of various FENE dumbbell models. J. Non-Newton. Fluid Mech. 68(1), 17–42 (1997)CrossRef

31.

Hyon, Y., Du, Q., Liu, C.: An enhanced macroscopic closure approximation to the micro-macro FENE model for polymeric materials. Multiscale Model. Simul. 7(2), 978–1002 (2008)CrossRefMATHMathSciNet

32.

Hesthaven, J., Warburton, T.: Nodal Discontinuous Galerkin Methods, Algorithms, Analysis and Applications. Springer, Berlin (2008)MATH

33.

Jourdain, B., Le Bris, C., Lelièvre, T., Otto, F.: Long time asymptotics of a multiscale model for polymeric fluid flows. Arch. Ration. Mech. Anal. 181(1), 97–148 (2006)CrossRefMATHMathSciNet

34.

Jourdain, B., Lelièvre, T.: Mathematical analysis of a stochastic differential equation arising in the micro-macro modelling of polymeric fluids. In: Probabilistic Methods in Fluids, pp. 205–223. World Sci. Publ., River Edge, NJ (2003)

35.

Jourdain, B., Lelièvre, T., Le Bris, C.: Existence of solution for a micro-macro model of polymeric fluid: the FENE model. J. Funct. Anal. 209(1), 162–193 (2004)CrossRefMATHMathSciNet

36.

Knezevic, D.J., Süli, E.: Spectral Galerkin approximation of Fokker–Planck equations with unbounded drift. M2AN Math. Model. Numer. Anal. 43(3), 445–485 (2009)CrossRefMATHMathSciNet

37.

Li, B.: Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer. Springer, London (2006)MATH

38.

Larsen, E.W., Levermore, C.D., Pomraning, G.C., Sanderson, J.G.: Discretization methods for one-dimensional Fokker–Planck operators. J. Comput. Phys. 61(3), 359–390 (1985)CrossRefMATHMathSciNet

39.

Le Bris, C., Lelièvre.: Micro-macro models for viscoelastic fluids: modeling, mathematics and numerics. arXiv:1102.0325v1[math-ph] (2011)

40.

Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal. 47(1), 675–698 (2009)CrossRefMATHMathSciNet

41.

Liu, H., Shin, J.: The Cauchy–Dirichlet problem for the FENE dumbbell model of polymeric flows. SIAM J. Math. Anal. 44(5), 3617–3648 (2012)CrossRefMATHMathSciNet

42.

Liu, H., Shin, J.: Global well-posedness for the microscopic FENE model with a sharp boundary condition. J. Differ. Equ. 252(1), 641–662 (2012)CrossRefMATHMathSciNet

43.

Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 8(3), 541–564 (2010)MathSciNet

44.

Liu, H., Yu, H.: An entropy satisfying conservative method for the Fokker–Planck equation of FENE dumbbell model for polymers. SIAM J. Numer. Anal. 50(3), 1207–1239 (2012)CrossRefMATHMathSciNet

45.

Liu, H., Yu, H.: Maximum-principle-satisfying third order discontinuous Galerkin schemes for Fokker–Planck equations. SIAM J. Sci. Comput. (2013). http://www.ams.org/amsmtgs/2216_abstracts/1097-65-286.pdf

46.

Lozinski, A., Chauvière, C.: A fast solver for Fokker–Planck equation applied to viscoelastic flows calculations: 2D FENE model. J. Comput. Phys. 189(2), 607–625 (2003)CrossRefMATHMathSciNet

47.

Masmoudi, N.: Well-posedness for the FENE dumbbell model of polymeric flows. Commun. Pure Appl. Math. 61(12), 1685–1714 (2008)CrossRefMATHMathSciNet

48.

Masmoudi, N.: Global existence of weak solutions to the FENE dumbbell model of polymeric flows. Invent. Math. 191(2), 427–500 (2013)CrossRefMATHMathSciNet

49.

Markowich, P.A., Villani, C.: On the trend to equilibrium for the Fokker–Planck equation: an interplay between physics and functional analysis. Phys. Funct. Anal. Mat. Contemp. (SBM) 19, 1–29 (1999)MathSciNet

50.

Oden, J.T., Babuška, I., Baumann, C.E.: A discontinuous \(hp\) finite element method for diffusion problems. J. Comput. Phys. 146(2), 491–519 (1998)CrossRefMATHMathSciNet

51.

Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, (1973)

52.

Riviére, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, Theory and Implementation, SIAM (2008)

53.

Samaey, G., Lelièvre, T., Legat, V.: A numerical closure approach for kinetic models of polymeric fluids: exploring closure relations for FENE dumbbells. Comput. Fluids 43, 119–133 (2011)CrossRefMATHMathSciNet

54.

Shu, C.-W.: Discontinuous Galerkin methods: general approach and stability. In Bertoluzza, S., Falletta, S., Russo, G., Shu, C.-W. (eds.) Numerical Solutions of Partial Differential Equations, pp. 149–201, Birkhauser Basel (2009)

55.

Shen, J., Yu, H.J.: On the approximation of the Fokker–Planck equation of FENE dumbbell model, I: a new weighted formulation and an optimal Spectral-Galerkin algorithm in 2-D. SIAM J. Numer. Anal. 50(3), 1136–1161 (2012)CrossRefMATHMathSciNet

56.

van Leer, B., Nomura, S.: Discontinuous Galerkin for diffusion. In: Proceedings of 17th AIAA Computational Fluid Dynamics Conference (6 June 2005), AIAA-2005-5108 (2005)

57.

Wang, H., Li, K., Zhang, P.: Crucial properties of the moment closure model FENE-QE. J. Non-Newton. Fluid Mech. 150(2–3), 80–92 (2008)CrossRefMATH

58.

Warner, H.R.: Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells. Ind. Eng. Chem. Fundam. 11(3), 379–387 (1972)CrossRefMathSciNet

59.

Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161 (1978)CrossRefMATHMathSciNet

60.

Zhang, H., Zhang, P.: Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal. 181(2), 373–400 (2006)CrossRefMATHMathSciNet

Title: The Entropy Satisfying Discontinuous Galerkin Method for Fokker–Planck equations
Authors: Hailiang Liu
Hui Yu
Publication date: 01-03-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2015
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-014-9878-1

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 3/2015

High-Order Flux Reconstruction Schemes with Minimal Dispersion and Dissipation

A Frequency-Domain Approach for the $$\hbox {P}_1$$ P 1 Approximation of Time-Dependent Radiative Transfer

Design of Loop’s Subdivision Surfaces by Fourth-Order Geometric PDEs with $$G^1$$ G 1 Boundary Conditions

Numerical Treatment of Interfaces for Second-Order Wave Equations

Convergence Analysis of a Minimax Method for Finding Multiple Solutions of Semilinear Elliptic Equation: Part I—On Polyhedral Domain

Runge–Kutta Residual Distribution Schemes

Premium Partner