January  2017, 37(1): 405-434. doi: 10.3934/dcds.2017017

Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations

Zentrum Mathematik, TU München Boltzmannstr. 3, D-85748 Garching, Germany

Received  January 2015 Revised  August 2016 Published  November 2016

Fund Project: This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”.

A fully discrete Lagrangian scheme for solving a family of fourth order equations numerically is presented. The discretization is based on the equations' underlying gradient flow structure with respect to the Wasserstein metric, and preserves numerous of their most important structural properties by construction, like conservation of mass and entropy-dissipation.

In this paper, the long-time behavior of our discretization is analysed: We show that discrete solutions decay exponentially to equilibrium at the same rate as smooth solutions of the original problem. Moreover, we give a proof of convergence of discrete entropy minimizers towards Barenblatt-profiles or Gaussians, respectively, using $Γ$-convergence.

Citation: Horst Osberger. Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 405-434. doi: 10.3934/dcds.2017017
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. doi: 978-3-7643-2428-5.

[2]

L. AmbrosioS. Lisini and G. Savaré, Stability of flows associated to gradient vector fields and convergence of iterated transport maps, Manuscripta Mathematica, 121 (2006), 1-50. doi: 10.1007/s00229-006-0003-0.

[3]

J. Becker and G. Grün, The thin-film equation: Recent advances and some new perspectives, Journal of Physics: Condensed Matter, 17 (2015), 291-307. doi: 10.1088/0953-8984/17/9/002.

[4]

F. Bernis and F. Avner, Higher order nonlinear degenerate parabolic equations, Journal of Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y.

[5]

M. BertschR. Dal PassoH. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Journal of Differential Equations, 3 (1998), 417-440.

[6]

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

[7]

P. M. BleherJ. L. Lebowitz and E. R. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Communications on Pure and Applied Mathematics, 47 (1994), 923-942. doi: 10.1002/cpa.3160470702.

[8]

A. Braides, $Γ$ -convergence for Beginners Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[9]

C. J. BuddG. J. CollinsW. Z. Huang and R. D. Russell, Self-similar numerical solutions of the porous-medium equation using moving mesh methods, The Royal Society of London. Philosophical Transactions. Series A. Mathematical, Physical and Engineering Sciences, 357 (1999), 1047-1077. doi: 10.1098/rsta.1999.0364.

[10]

M. BukalE. Emmrich and A. Jüngel, Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation, Numerische Mathematik, 127 (2014), 365-396. doi: 10.1007/s00211-013-0588-7.

[11]

M. BurgerJ. A. Carrillo and M.-T. Wolfram, A mixed finite element method for nonlinear diffusion equations, Kinetic and Related Models, 3 (2010), 59-83. doi: 10.3934/krm.2010.3.59.

[12]

M. J. CáceresJ. A. Carrillo and G. Toscani, Long-time behavior for a nonlinear fourth-order parabolic equation, Transactions of the American Mathematical Society, 357 (2005), 1161-1175. doi: 10.1090/S0002-9947-04-03528-7.

[13]

E. A. Carlen and S. Ulusoy, Asymptotic equipartition and long time behavior of solutions of a thin-film equation, Journal of Differential Equations, 241 (2007), 279-292. doi: 10.1016/j.jde.2007.07.005.

[14]

J. A. CarrilloJ. DolbeaultI. Gentil and A. Jüngel, Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 6 (2006), 1027-1050. doi: 10.3934/dcdsb.2006.6.1027.

[15]

J. A. CarrilloA. JüngelP. A. MarkowichG. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized {S}obolev inequalities, Monatshefte für Mathematik, 133 (2001), 1-82. doi: 10.1007/s006050170032.

[16]

J. A. CarrilloA. Jüngel and S. Tang, Positive entropic schemes for a nonlinear fourth-order parabolic equation, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 3 (2003), 1-20.

[17]

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 (2009/10), 4305-4329. doi: 10.1137/080739574.

[18]

J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin film equation, Communications in Mathematical Physics, 225 (2002), 551-571. doi: 10.1007/s002200100591.

[19]

J. A. Carrillo and M. -T. Wolfram, A finite element method for nonlinear continuity equations in Lagrangian coordinates, work in progress.

[20]

F. Cavalli and G. Naldi, A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation, Kinetic and Related Models, 3 (2010), 123-142. doi: 10.3934/krm.2010.3.123.

[21]

R. Dal PassoH. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM Journal on Mathematical Analysis, 29 (1998), 321-342. doi: 10.1137/S0036141096306170.

[22]

J. Denzler and R. J. McCann, Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, & nonradial focusing geometries, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, 25 (2008), 865-888. doi: 10.1016/j.anihpc.2007.05.002.

[23]

B. DerridaJ. L. LebowitzE. R. Speer and H. Spohn, Dynamics of an anchored Toom interface, Journal of Physics. A. Mathematical and General, 24 (1991), 4805-4834. doi: 10.1088/0305-4470/24/20/015.

[24]

B. DerridaJ. L. LebowitzE. R. Speer and H. Spohn, Fluctuations of a stationary nonequilibrium interface, Physical Review Letters, 67 (1991), 165-168. doi: 10.1103/PhysRevLett.67.165.

[25]

B. DüringD. Matthes and J. P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 14 (2010), 935-959. doi: 10.3934/dcdsb.2010.14.935.

[26]

L. C. EvansO. Savin and W. Gangbo, Diffeomorphisms and nonlinear heat flows, SIAM Journal on Mathematical Analysis, 37 (2005), 737-751. doi: 10.1137/04061386X.

[27]

J. Fischer, Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift-diffusion models, SIAM Journal on Mathematical Analysis, 38 (2013), 2004-2047. doi: 10.1080/03605302.2013.823548.

[28]

L. Giacomelli and F. Otto, Variational 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.

[29]

U. GianazzaG. Savaré and G. Toscani, The {W}asserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Archive for Rational Mechanics and Analysis, 194 (2009), 133-220. doi: 10.1007/s00205-008-0186-5.

[30]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation Monographs in Mathematics, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.

[31]

L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM Journal on Numerical Analysis, 43 (2006), 2590-2606. doi: 10.1137/040608672.

[32]

L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM Journal on Scientific Computing, 28 (2006), 1203-1227. doi: 10.1137/050628015.

[33]

G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Communications in Partial Differential Equations, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193.

[34]

M. P. GualdaniA. Jüngel and G. Toscani, A nonlinear fourth-order parabolic equation with nonhomogeneous boundary conditions, SIAM Journal on Mathematical Analysis, 37 (2006), 1761-1779. doi: 10.1137/S0036141004444615.

[35]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[36]

A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659. doi: 10.1088/0951-7715/19/3/006.

[37]

A. Jüngel and D. Matthes, The Derrida-Lebowitz-Speer-Spohn equation: Existence, nonuniqueness, and decay rates of the solutions, SIAM Journal on Mathematical Analysis, 39 (2008), 1996-2015. doi: 10.1137/060676878.

[38]

A. Jüngel and R. Pinnau, Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems, SIAM Journal on Mathematical Analysis, 32 (2000), 760-777. doi: 10.1137/S0036141099360269.

[39]

A. Jüngel and R. Pinnau, A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system, SIAM Journal on Numerical Analysis, 39 (2001), 385-406. doi: 10.1137/S0036142900369362.

[40]

A. Jüngel and G. Toscani, Exponential time decay of solutions to a nonlinear fourth-order parabolic equation, Journal of Applied Mathematics and Physics, 54 (2003), 377-386. doi: 10.1007/s00033-003-1026-y.

[41]

A. Jüngel and I. Violet, First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 8 (2007), 861-877. doi: 10.3934/dcdsb.2007.8.861.

[42]

D. Kinderlehrer and N. J. Walkington, Approximation of parabolic equations using the Wasserstein metric, M2AN. Mathematical Modelling and Numerical Analysis, 33 (1999), 837-852. doi: 10.1051/m2an:1999166.

[43]

R. C. MacCamy and E. Socolovsky, A numerical procedure for the porous media equation, Computers & Mathematics with Applications. An International Journal, 11 (1985), 315-319. doi: 10.1016/0898-1221(85)90156-7.

[44]

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

[45]

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.

[46]

D. Matthes and H. Osberger, A convergent Lagrangian discretization for a nonlinear fourth-order equation, Foundations of Computational Mathematics, 17 (2015), 1-54. doi: 10.1007/s10208-015-9284-6.

[47]

A. OronS. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, American Physical Society, 69 (1997), 931-980. doi: 10.1103/RevModPhys.69.931.

[48]

G. Russo, Deterministic diffusion of particles, Communications on Pure and Applied Mathematics, 43 (1990), 697-733. doi: 10.1002/cpa.3160430602.

[49]

C. Villani, Topics in Optimal Transportation American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. doi: 978-3-7643-2428-5.

[2]

L. AmbrosioS. Lisini and G. Savaré, Stability of flows associated to gradient vector fields and convergence of iterated transport maps, Manuscripta Mathematica, 121 (2006), 1-50. doi: 10.1007/s00229-006-0003-0.

[3]

J. Becker and G. Grün, The thin-film equation: Recent advances and some new perspectives, Journal of Physics: Condensed Matter, 17 (2015), 291-307. doi: 10.1088/0953-8984/17/9/002.

[4]

F. Bernis and F. Avner, Higher order nonlinear degenerate parabolic equations, Journal of Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y.

[5]

M. BertschR. Dal PassoH. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Journal of Differential Equations, 3 (1998), 417-440.

[6]

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

[7]

P. M. BleherJ. L. Lebowitz and E. R. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Communications on Pure and Applied Mathematics, 47 (1994), 923-942. doi: 10.1002/cpa.3160470702.

[8]

A. Braides, $Γ$ -convergence for Beginners Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[9]

C. J. BuddG. J. CollinsW. Z. Huang and R. D. Russell, Self-similar numerical solutions of the porous-medium equation using moving mesh methods, The Royal Society of London. Philosophical Transactions. Series A. Mathematical, Physical and Engineering Sciences, 357 (1999), 1047-1077. doi: 10.1098/rsta.1999.0364.

[10]

M. BukalE. Emmrich and A. Jüngel, Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation, Numerische Mathematik, 127 (2014), 365-396. doi: 10.1007/s00211-013-0588-7.

[11]

M. BurgerJ. A. Carrillo and M.-T. Wolfram, A mixed finite element method for nonlinear diffusion equations, Kinetic and Related Models, 3 (2010), 59-83. doi: 10.3934/krm.2010.3.59.

[12]

M. J. CáceresJ. A. Carrillo and G. Toscani, Long-time behavior for a nonlinear fourth-order parabolic equation, Transactions of the American Mathematical Society, 357 (2005), 1161-1175. doi: 10.1090/S0002-9947-04-03528-7.

[13]

E. A. Carlen and S. Ulusoy, Asymptotic equipartition and long time behavior of solutions of a thin-film equation, Journal of Differential Equations, 241 (2007), 279-292. doi: 10.1016/j.jde.2007.07.005.

[14]

J. A. CarrilloJ. DolbeaultI. Gentil and A. Jüngel, Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 6 (2006), 1027-1050. doi: 10.3934/dcdsb.2006.6.1027.

[15]

J. A. CarrilloA. JüngelP. A. MarkowichG. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized {S}obolev inequalities, Monatshefte für Mathematik, 133 (2001), 1-82. doi: 10.1007/s006050170032.

[16]

J. A. CarrilloA. Jüngel and S. Tang, Positive entropic schemes for a nonlinear fourth-order parabolic equation, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 3 (2003), 1-20.

[17]

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 (2009/10), 4305-4329. doi: 10.1137/080739574.

[18]

J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin film equation, Communications in Mathematical Physics, 225 (2002), 551-571. doi: 10.1007/s002200100591.

[19]

J. A. Carrillo and M. -T. Wolfram, A finite element method for nonlinear continuity equations in Lagrangian coordinates, work in progress.

[20]

F. Cavalli and G. Naldi, A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation, Kinetic and Related Models, 3 (2010), 123-142. doi: 10.3934/krm.2010.3.123.

[21]

R. Dal PassoH. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM Journal on Mathematical Analysis, 29 (1998), 321-342. doi: 10.1137/S0036141096306170.

[22]

J. Denzler and R. J. McCann, Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, & nonradial focusing geometries, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, 25 (2008), 865-888. doi: 10.1016/j.anihpc.2007.05.002.

[23]

B. DerridaJ. L. LebowitzE. R. Speer and H. Spohn, Dynamics of an anchored Toom interface, Journal of Physics. A. Mathematical and General, 24 (1991), 4805-4834. doi: 10.1088/0305-4470/24/20/015.

[24]

B. DerridaJ. L. LebowitzE. R. Speer and H. Spohn, Fluctuations of a stationary nonequilibrium interface, Physical Review Letters, 67 (1991), 165-168. doi: 10.1103/PhysRevLett.67.165.

[25]

B. DüringD. Matthes and J. P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 14 (2010), 935-959. doi: 10.3934/dcdsb.2010.14.935.

[26]

L. C. EvansO. Savin and W. Gangbo, Diffeomorphisms and nonlinear heat flows, SIAM Journal on Mathematical Analysis, 37 (2005), 737-751. doi: 10.1137/04061386X.

[27]

J. Fischer, Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift-diffusion models, SIAM Journal on Mathematical Analysis, 38 (2013), 2004-2047. doi: 10.1080/03605302.2013.823548.

[28]

L. Giacomelli and F. Otto, Variational 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.

[29]

U. GianazzaG. Savaré and G. Toscani, The {W}asserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Archive for Rational Mechanics and Analysis, 194 (2009), 133-220. doi: 10.1007/s00205-008-0186-5.

[30]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation Monographs in Mathematics, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.

[31]

L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM Journal on Numerical Analysis, 43 (2006), 2590-2606. doi: 10.1137/040608672.

[32]

L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM Journal on Scientific Computing, 28 (2006), 1203-1227. doi: 10.1137/050628015.

[33]

G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Communications in Partial Differential Equations, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193.

[34]

M. P. GualdaniA. Jüngel and G. Toscani, A nonlinear fourth-order parabolic equation with nonhomogeneous boundary conditions, SIAM Journal on Mathematical Analysis, 37 (2006), 1761-1779. doi: 10.1137/S0036141004444615.

[35]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[36]

A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659. doi: 10.1088/0951-7715/19/3/006.

[37]

A. Jüngel and D. Matthes, The Derrida-Lebowitz-Speer-Spohn equation: Existence, nonuniqueness, and decay rates of the solutions, SIAM Journal on Mathematical Analysis, 39 (2008), 1996-2015. doi: 10.1137/060676878.

[38]

A. Jüngel and R. Pinnau, Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems, SIAM Journal on Mathematical Analysis, 32 (2000), 760-777. doi: 10.1137/S0036141099360269.

[39]

A. Jüngel and R. Pinnau, A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system, SIAM Journal on Numerical Analysis, 39 (2001), 385-406. doi: 10.1137/S0036142900369362.

[40]

A. Jüngel and G. Toscani, Exponential time decay of solutions to a nonlinear fourth-order parabolic equation, Journal of Applied Mathematics and Physics, 54 (2003), 377-386. doi: 10.1007/s00033-003-1026-y.

[41]

A. Jüngel and I. Violet, First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 8 (2007), 861-877. doi: 10.3934/dcdsb.2007.8.861.

[42]

D. Kinderlehrer and N. J. Walkington, Approximation of parabolic equations using the Wasserstein metric, M2AN. Mathematical Modelling and Numerical Analysis, 33 (1999), 837-852. doi: 10.1051/m2an:1999166.

[43]

R. C. MacCamy and E. Socolovsky, A numerical procedure for the porous media equation, Computers & Mathematics with Applications. An International Journal, 11 (1985), 315-319. doi: 10.1016/0898-1221(85)90156-7.

[44]

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

[45]

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.

[46]

D. Matthes and H. Osberger, A convergent Lagrangian discretization for a nonlinear fourth-order equation, Foundations of Computational Mathematics, 17 (2015), 1-54. doi: 10.1007/s10208-015-9284-6.

[47]

A. OronS. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, American Physical Society, 69 (1997), 931-980. doi: 10.1103/RevModPhys.69.931.

[48]

G. Russo, Deterministic diffusion of particles, Communications on Pure and Applied Mathematics, 43 (1990), 697-733. doi: 10.1002/cpa.3160430602.

[49]

C. Villani, Topics in Optimal Transportation American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.

Figure 2.  Evolution of a discrete solution $u_\Delta$, evaluated at different times $t = 0,0.05,0.1,0.15,0.175,0.25$ (from top left to bottom right). The red line is the corresponding Barenblatt-profile ${{\text{b}}_{\alpha ,\lambda }}$.
Figure 1.  Left: Numerically observed decay of $H_{\alpha,\lambda}(t)-{\mathbf{H}_{\alpha ,\lambda }^{\min }}$ and $F_{\alpha,\lambda}(t)-{\mathbf{F}_{\alpha ,\lambda }^{\min }}$ along a time period of $t\in[0,0.8]$, using $K=25,50,100,200$, in comparison to the upper bounds $({\mathcal{H}_{\alpha ,\lambda }}(u^0)-{\mathcal{H}_{\alpha ,\lambda }}({{\text{b}}_{\alpha ,\lambda }}))\exp(-2\lambda t)$ and $({\mathcal{F}_{\alpha ,\lambda }}(u^0)-{\mathcal{F}_{\alpha ,\lambda }}({{\text{b}}_{\alpha ,\lambda }}))\exp(-2\lambda t)$, respectively. Right: Convergence of discrete minimizers $u_{\delta }^{\min }$ with a rate of $K^{-1.5}$.
Figure 3.  Snapshots of the densities $\text{b}_{\alpha ,0}^{*}(t,\cdot)$ (red lines) and $u_\Delta$ (black lines) for the initial condition $\text{b}_{\alpha ,0}^{*}(0,\cdot)$ at times $t=0$ and $t= 0.1\cdot 10^{i}$, $i=0,\ldots,3$, using $K=50$ grid points and the time step size $\tau=10^{-3}$.
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