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

Previous Article
Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations
DCDS Home
This Issue
Next Article
A new proof of continuity of Lyapunov exponents for a class of $ C^2 $ quasiperiodic Schrödinger cocycles without LDT

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 - A, 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	E. Giusti, Minimal Surfaces and Functions of Bounded Variation, vol. 80, Birkhauser Verlag, 1984. doi: 10.1007/978-1-4684-9486-0.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[31]	G. Peyré, Entropic approximation of Wasserstein gradient flows, SIAM Journal on Imaging Sciences, 8 (2015), 2323-2351. doi: 10.1137/15M1010087.
[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.
[33]	F. Santambrogio, Optimal transport for Applied Mathematicians, Springer, 2015. doi: 10.1007/978-3-319-20828-2.
[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.
[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.
[36]	C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 2003. doi: 10.1007/b12016.
[37]	C. Villani, Optimal Transport: Old and New, vol. 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	E. Giusti, Minimal Surfaces and Functions of Bounded Variation, vol. 80, Birkhauser Verlag, 1984. doi: 10.1007/978-1-4684-9486-0.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[31]	G. Peyré, Entropic approximation of Wasserstein gradient flows, SIAM Journal on Imaging Sciences, 8 (2015), 2323-2351. doi: 10.1137/15M1010087.
[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.
[33]	F. Santambrogio, Optimal transport for Applied Mathematicians, Springer, 2015. doi: 10.1007/978-3-319-20828-2.
[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.
[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.
[36]	C. Villani, Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 2003. doi: 10.1007/b12016.
[37]	C. Villani, Optimal Transport: Old and New, vol. 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.
[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.
[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.
[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.

[1]	Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841
[2]	Bertram Düring, Daniel Matthes, Josipa Pina Milišić. A gradient flow scheme for nonlinear fourth order equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 935-959. doi: 10.3934/dcdsb.2010.14.935
[3]	Christoph Walker. Age-dependent equations with non-linear diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 691-712. doi: 10.3934/dcds.2010.26.691
[4]	Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic & Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713
[5]	Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : ⅰ-ⅳ. doi: 10.3934/dcdss.201702i
[6]	Wei Qu, Siu-Long Lei, Seak-Weng Vong. A note on the stability of a second order finite difference scheme for space fractional diffusion equations. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 317-325. doi: 10.3934/naco.2014.4.317
[7]	Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems & Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004
[8]	Michela Procesi. Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 541-552. doi: 10.3934/dcds.2005.13.541
[9]	Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393
[10]	Antonin Chambolle, Francesco Doveri. Minimizing movements of the Mumford and Shah energy. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 153-174. doi: 10.3934/dcds.1997.3.153
[11]	Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[12]	Simona Fornaro, Ugo Gianazza. Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 481-492. doi: 10.3934/dcds.2010.26.481
[13]	Dmitry Dolgopyat. Bouncing balls in non-linear potentials. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 165-182. doi: 10.3934/dcds.2008.22.165
[14]	Dorin Ervin Dutkay and Palle E. T. Jorgensen. Wavelet constructions in non-linear dynamics. Electronic Research Announcements, 2005, 11: 21-33.
[15]	Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems & Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675
[16]	Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435
[17]	Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239
[18]	Wenbin Chen, Wenqiang Feng, Yuan Liu, Cheng Wang, Steven M. Wise. A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 149-182. doi: 10.3934/dcdsb.2018090
[19]	Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517
[20]	Sanjay Khattri. Another note on some quadrature based three-step iterative methods for non-linear equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 549-555. doi: 10.3934/naco.2013.3.549

American Institute of Mathematical Sciences