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May  2019, 39(5): 2893-2913. doi: 10.3934/dcds.2019120

A BDF2-approach for the non-linear Fokker-Planck equation

Simon Plazotta

Boltzmannstra. 3, D-85747 Garching, Germany

Received  July 2018 Revised  October 2018 Published  January 2019

Fund Project: This research has been supported by the German Research Foundation (DFG), SFB TRR 109. The author would like to thank Daniel Matthes for helpful discussions and remarks.

We prove convergence of a variational formulation of the BDF2 method applied to the non-linear Fokker-Planck equation. Our approach is inspired by the JKO-method and exploits the differential structure of the underlying $ L^2 $-Wasserstein space. The technique presented here extends and strengthens the results of our own recent work [27] on the BDF2 method for general metric gradient flows in the special case of the non-linear Fokker-Planck equation: firstly, we do not require uniform semi-convexity of the augmented energy functional; secondly, we prove strong instead of merely weak convergence of the time-discrete approximations; thirdly, we directly prove without using the abstract theory of curves of maximal slope that the obtained limit curve is a weak solution of the non-linear Fokker-Planck equation.

Keywords: Gradient flow, second order scheme, BDF2, minimizing movements, non-linear diffusion equations.
Mathematics Subject Classification: Primary: 34G25, 35A15, 35G25, 35K46, 65L06, 65J08.
Citation: Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, vol. 254, Clarendon Press Oxford, 2000.  Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[3]

J.-D. Benamou, G. Carlier, Q. Mérigot and E. Oudet, Discretization of functionals involving the monge–ampère operator, Numerische Mathematik, 134 (2016), 611-636.  doi: 10.1007/s00211-015-0781-y.  Google Scholar

[4]

A. Blanchet and P. Laurençot, The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\mathbb{R}^d,\ d\ge3$, Comm. Partial Differential Equations, 38 (2013), 658-686.  doi: 10.1080/03605302.2012.757705.  Google Scholar

[5]

A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak–Keller–Segel model, SIAM Journal on Numerical Analysis, 46 (2008), 691-721.  doi: 10.1137/070683337.  Google Scholar

[6]

V. Calvez and T. O. Gallouët, Blow-up phenomena for gradient flows of discrete homogeneous functionals, Applied Mathematics & Optimization, (2017), 1–29. doi: 10.1007/s00245-017-9443-z.  Google Scholar

[7]

J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.  doi: 10.1215/00127094-2010-211.  Google Scholar

[8]

J. Carrillo, F. Patacchini, P. Sternberg and G. Wolansky, Convergence of a particle method for diffusive gradient flows in one dimension, SIAM Journal on Mathematical Analysis, 48 (2016), 3708-3741.  doi: 10.1137/16M1077210.  Google Scholar

[9]

J. A. Carrillo, H. Ranetbauer and M.-T. Wolfram, Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms, Journal of Computational Physics, 327 (2016), 186-202, URL http://www.sciencedirect.com/science/article/pii/S0021999116304612. doi: 10.1016/j.jcp.2016.09.040.  Google Scholar

[10]

J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM Journal on Scientific Computing, 31 (2009), 4305-4329.  doi: 10.1137/080739574.  Google Scholar

[11]

M. Di Francesco, A. Esposito and S. Fagioli, Nonlinear degenerate cross-diffusion systems with nonlocal interaction, Nonlinear Analysis, 169 (2018), 94-117.  doi: 10.1016/j.na.2017.12.003.  Google Scholar

[12]

B. Düring, D. Matthes and J. P. Milišic, A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 935-959.  doi: 10.3934/dcdsb.2010.14.935.  Google Scholar

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M. Erbar et al., The heat equation on manifolds as a gradient flow in the Wasserstein space, in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, vol. 46, Institut Henri Poincaré, 2010, 1–23. doi: 10.1214/08-AIHP306.  Google Scholar

[14]

T. O. Gallouët and Q. Mérigot, A Lagrangian scheme à la Brenier for the incompressible Euler Equations, Foundations of Computational Mathematics, 18 (2018), 835-865.  doi: 10.1007/s10208-017-9355-y.  Google Scholar

[15]

L. Giacomelli and F. Otto, Variatonal formulation for the lubrication approximation of the Hele-Shaw flow, Calculus of Variations and Partial Differential Equations, 13 (2001), 377-403.  doi: 10.1007/s005260000077.  Google Scholar

[16]

U. Gianazza, G. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., 194 (2009), 133-220.  doi: 10.1007/s00205-008-0186-5.  Google Scholar

[17]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, vol. 80, Birkhauser Verlag, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[18]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.  Google Scholar

[19]

O. Junge, D. Matthes and H. Osberger, A fully discrete variational scheme for solving nonlinear fokker–planck equations in multiple space dimensions, SIAM Journal on Numerical Analysis, 55 (2017), 419-443.  doi: 10.1137/16M1056560.  Google Scholar

[20]

D. Kinderlehrer, L. Monsaingeon and X. Xu, A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 137-164.  doi: 10.1051/cocv/2015043.  Google Scholar

[21]

L. Laguzet, High order variational numerical schemes with application to Nash -MFG vaccination games, Ric. Mat., 67 (2018), 247-269.  doi: 10.1007/s11587-018-0366-z.  Google Scholar

[22]

P. Laurençot and B.-V. Matioc, A gradient flow approach to a thin film approximation of the Muskat problem, Calc. Var. Partial Differential Equations, 47 (2013), 319-341.  doi: 10.1007/s00526-012-0520-5.  Google Scholar

[23]

G. Legendre and G. Turinici, Second-order in time schemes for gradient flows in Wasserstein and geodesic metric spaces, Comptes Rendus Mathematique, 355 (2017), 345-353, URL http://www.sciencedirect.com/science/article/pii/S1631073X17300365. doi: 10.1016/j.crma.2017.02.001.  Google Scholar

[24]

S. Lisini, D. Matthes and G. Savaré, Cahn-Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics, J. Differential Equations, 253 (2012), 814-850.  doi: 10.1016/j.jde.2012.04.004.  Google Scholar

[25]

D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397.  doi: 10.1080/03605300903296256.  Google Scholar

[26]

D. Matthes and H. Osberger, Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 697-726.  doi: 10.1051/m2an/2013126.  Google Scholar

[27]

D. Matthes and S. Plazotta, A variational formulation of the BDF2 method for metric gradient flows, to appear in ESAIM: Mathematical Modelling and Numerical Analysis. doi: 10.1051/m2an/2018045.  Google Scholar

[28]

D. Matthes and B. Söllner, Convergent Lagrangian discretization for drift-diffusion with nonlocal aggregation, in Innovative Algorithms and Analysis, Springer, 16 (2017), 313–351. doi: 10.1007/978-3-319-49262-9_12.  Google Scholar

[29]

D. Matthes and J. Zinsl, Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type, Nonlinear Analysis, 159 (2017), 316-338.  doi: 10.1016/j.na.2016.12.002.  Google Scholar

[30]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.  Google Scholar

[31]

G. Peyré, Entropic approximation of Wasserstein gradient flows, SIAM Journal on Imaging Sciences, 8 (2015), 2323-2351.  doi: 10.1137/15M1010087.  Google Scholar

[32]

R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie V, 2 (2003), 395-431.   Google Scholar

[33]

F. Santambrogio, Optimal transport for Applied Mathematicians, Springer, 2015. doi: 10.1007/978-3-319-20828-2.  Google Scholar

[34]

K.-T. Sturm, Convex functionals of probability measures and nonlinear diffusions on manifolds, Journal de mathématiques pures et appliquées, 84 (2005), 149–168. doi: 10.1016/j.matpur.2004.11.002.  Google Scholar

[35]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bulletin of Mathematical Biology, 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[36]

C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 2003. doi: 10.1007/b12016.  Google Scholar

[37]

C. Villani, Optimal Transport: Old and New, vol. 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[38]

M. Westdickenberg and J. Wilkening, Variational particle schemes for the porous medium equation and for the system of isentropic euler equations, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 133-166.  doi: 10.1051/m2an/2009043.  Google Scholar

[39]

J. Zinsl and D. Matthes, Exponential convergence to equilibrium in a coupled gradient flow system modeling chemotaxis, Analysis & PDE, 8 (2015), 425-466.  doi: 10.2140/apde.2015.8.425.  Google Scholar

[40]

J. Zinsl and D. Matthes, Transport distances and geodesic convexity for systems of degenerate diffusion equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 3397-3438.  doi: 10.1007/s00526-015-0909-z.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, vol. 254, Clarendon Press Oxford, 2000.  Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[3]

J.-D. Benamou, G. Carlier, Q. Mérigot and E. Oudet, Discretization of functionals involving the monge–ampère operator, Numerische Mathematik, 134 (2016), 611-636.  doi: 10.1007/s00211-015-0781-y.  Google Scholar

[4]

A. Blanchet and P. Laurençot, The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\mathbb{R}^d,\ d\ge3$, Comm. Partial Differential Equations, 38 (2013), 658-686.  doi: 10.1080/03605302.2012.757705.  Google Scholar

[5]

A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak–Keller–Segel model, SIAM Journal on Numerical Analysis, 46 (2008), 691-721.  doi: 10.1137/070683337.  Google Scholar

[6]

V. Calvez and T. O. Gallouët, Blow-up phenomena for gradient flows of discrete homogeneous functionals, Applied Mathematics & Optimization, (2017), 1–29. doi: 10.1007/s00245-017-9443-z.  Google Scholar

[7]

J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.  doi: 10.1215/00127094-2010-211.  Google Scholar

[8]

J. Carrillo, F. Patacchini, P. Sternberg and G. Wolansky, Convergence of a particle method for diffusive gradient flows in one dimension, SIAM Journal on Mathematical Analysis, 48 (2016), 3708-3741.  doi: 10.1137/16M1077210.  Google Scholar

[9]

J. A. Carrillo, H. Ranetbauer and M.-T. Wolfram, Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms, Journal of Computational Physics, 327 (2016), 186-202, URL http://www.sciencedirect.com/science/article/pii/S0021999116304612. doi: 10.1016/j.jcp.2016.09.040.  Google Scholar

[10]

J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM Journal on Scientific Computing, 31 (2009), 4305-4329.  doi: 10.1137/080739574.  Google Scholar

[11]

M. Di Francesco, A. Esposito and S. Fagioli, Nonlinear degenerate cross-diffusion systems with nonlocal interaction, Nonlinear Analysis, 169 (2018), 94-117.  doi: 10.1016/j.na.2017.12.003.  Google Scholar

[12]

B. Düring, D. Matthes and J. P. Milišic, A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 935-959.  doi: 10.3934/dcdsb.2010.14.935.  Google Scholar

[13]

M. Erbar et al., The heat equation on manifolds as a gradient flow in the Wasserstein space, in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, vol. 46, Institut Henri Poincaré, 2010, 1–23. doi: 10.1214/08-AIHP306.  Google Scholar

[14]

T. O. Gallouët and Q. Mérigot, A Lagrangian scheme à la Brenier for the incompressible Euler Equations, Foundations of Computational Mathematics, 18 (2018), 835-865.  doi: 10.1007/s10208-017-9355-y.  Google Scholar

[15]

L. Giacomelli and F. Otto, Variatonal formulation for the lubrication approximation of the Hele-Shaw flow, Calculus of Variations and Partial Differential Equations, 13 (2001), 377-403.  doi: 10.1007/s005260000077.  Google Scholar

[16]

U. Gianazza, G. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., 194 (2009), 133-220.  doi: 10.1007/s00205-008-0186-5.  Google Scholar

[17]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, vol. 80, Birkhauser Verlag, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[18]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.  Google Scholar

[19]

O. Junge, D. Matthes and H. Osberger, A fully discrete variational scheme for solving nonlinear fokker–planck equations in multiple space dimensions, SIAM Journal on Numerical Analysis, 55 (2017), 419-443.  doi: 10.1137/16M1056560.  Google Scholar

[20]

D. Kinderlehrer, L. Monsaingeon and X. Xu, A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 137-164.  doi: 10.1051/cocv/2015043.  Google Scholar

[21]

L. Laguzet, High order variational numerical schemes with application to Nash -MFG vaccination games, Ric. Mat., 67 (2018), 247-269.  doi: 10.1007/s11587-018-0366-z.  Google Scholar

[22]

P. Laurençot and B.-V. Matioc, A gradient flow approach to a thin film approximation of the Muskat problem, Calc. Var. Partial Differential Equations, 47 (2013), 319-341.  doi: 10.1007/s00526-012-0520-5.  Google Scholar

[23]

G. Legendre and G. Turinici, Second-order in time schemes for gradient flows in Wasserstein and geodesic metric spaces, Comptes Rendus Mathematique, 355 (2017), 345-353, URL http://www.sciencedirect.com/science/article/pii/S1631073X17300365. doi: 10.1016/j.crma.2017.02.001.  Google Scholar

[24]

S. Lisini, D. Matthes and G. Savaré, Cahn-Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics, J. Differential Equations, 253 (2012), 814-850.  doi: 10.1016/j.jde.2012.04.004.  Google Scholar

[25]

D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397.  doi: 10.1080/03605300903296256.  Google Scholar

[26]

D. Matthes and H. Osberger, Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 697-726.  doi: 10.1051/m2an/2013126.  Google Scholar

[27]

D. Matthes and S. Plazotta, A variational formulation of the BDF2 method for metric gradient flows, to appear in ESAIM: Mathematical Modelling and Numerical Analysis. doi: 10.1051/m2an/2018045.  Google Scholar

[28]

D. Matthes and B. Söllner, Convergent Lagrangian discretization for drift-diffusion with nonlocal aggregation, in Innovative Algorithms and Analysis, Springer, 16 (2017), 313–351. doi: 10.1007/978-3-319-49262-9_12.  Google Scholar

[29]

D. Matthes and J. Zinsl, Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type, Nonlinear Analysis, 159 (2017), 316-338.  doi: 10.1016/j.na.2016.12.002.  Google Scholar

[30]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.  Google Scholar

[31]

G. Peyré, Entropic approximation of Wasserstein gradient flows, SIAM Journal on Imaging Sciences, 8 (2015), 2323-2351.  doi: 10.1137/15M1010087.  Google Scholar

[32]

R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie V, 2 (2003), 395-431.   Google Scholar

[33]

F. Santambrogio, Optimal transport for Applied Mathematicians, Springer, 2015. doi: 10.1007/978-3-319-20828-2.  Google Scholar

[34]

K.-T. Sturm, Convex functionals of probability measures and nonlinear diffusions on manifolds, Journal de mathématiques pures et appliquées, 84 (2005), 149–168. doi: 10.1016/j.matpur.2004.11.002.  Google Scholar

[35]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bulletin of Mathematical Biology, 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[36]

C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 2003. doi: 10.1007/b12016.  Google Scholar

[37]

C. Villani, Optimal Transport: Old and New, vol. 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[38]

M. Westdickenberg and J. Wilkening, Variational particle schemes for the porous medium equation and for the system of isentropic euler equations, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 133-166.  doi: 10.1051/m2an/2009043.  Google Scholar

[39]

J. Zinsl and D. Matthes, Exponential convergence to equilibrium in a coupled gradient flow system modeling chemotaxis, Analysis & PDE, 8 (2015), 425-466.  doi: 10.2140/apde.2015.8.425.  Google Scholar

[40]

J. Zinsl and D. Matthes, Transport distances and geodesic convexity for systems of degenerate diffusion equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 3397-3438.  doi: 10.1007/s00526-015-0909-z.  Google Scholar

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