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.
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References
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)
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)
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)
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)
Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)
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)
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)
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)
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)
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)
Barrett, J.W., Süli, E.: Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers. ESAIM M2AN 46, 949–978 (2012)
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)
Chang, J.S., Cooper, G.: A practical difference scheme for Fokker–Planck equations. J. Comput. Phys. 6(1), 1–16 (1970)
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)
Chauvière, C., Lozinski, A.: Simulation of dilute polymer solutions using a Fokker–Planck equation. J. Comput. Fluids 33(5–6), 687–696 (2004)
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)
Chupin, L.: The FENE model for viscoelastic thin film flows. Methods Appl. Anal. 16(2), 217–261 (2009)
Chupin, L.: The FENE viscoelastic model and thin film flows. C. R. Math. Acad. Sci. Paris 347(17–18), 1041–1046 (2009)
Chupin, L.: Fokker–Planck equation in bounded domain. Ann. Inst. Fourier (Grenoble) 60(1), 217–255 (2010)
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)
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)
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)
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)
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)
Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)
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)
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)
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)
Herrchen, M., Öttinger, H.C.: A detailed comparison of various FENE dumbbell models. J. Non-Newton. Fluid Mech. 68(1), 17–42 (1997)
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)
Hesthaven, J., Warburton, T.: Nodal Discontinuous Galerkin Methods, Algorithms, Analysis and Applications. Springer, Berlin (2008)
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)
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)
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)
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)
Li, B.: Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer. Springer, London (2006)
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)
Le Bris, C., Lelièvre.: Micro-macro models for viscoelastic fluids: modeling, mathematics and numerics. arXiv:1102.0325v1[math-ph] (2011)
Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal. 47(1), 675–698 (2009)
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)
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)
Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 8(3), 541–564 (2010)
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)
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
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)
Masmoudi, N.: Well-posedness for the FENE dumbbell model of polymeric flows. Commun. Pure Appl. Math. 61(12), 1685–1714 (2008)
Masmoudi, N.: Global existence of weak solutions to the FENE dumbbell model of polymeric flows. Invent. Math. 191(2), 427–500 (2013)
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)
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)
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)
Riviére, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, Theory and Implementation, SIAM (2008)
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)
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)
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)
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)
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)
Warner, H.R.: Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells. Ind. Eng. Chem. Fundam. 11(3), 379–387 (1972)
Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161 (1978)
Zhang, H., Zhang, P.: Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal. 181(2), 373–400 (2006)
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This research was partially supported by the National Science Foundation under Grant DMS09-07963 and DMS13-12636
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Liu, H., Yu, H. The Entropy Satisfying Discontinuous Galerkin Method for Fokker–Planck equations. J Sci Comput 62, 803–830 (2015). https://doi.org/10.1007/s10915-014-9878-1
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DOI: https://doi.org/10.1007/s10915-014-9878-1