September  2020, 19(9): 4227-4256. doi: 10.3934/cpaa.2020190

A convergent Lagrangian discretization for $ p $-Wasserstein and flux-limited diffusion equations

Department of Mathematics, Technical University of Munich, 85747 Garching, Germany

*Corresponding author

Received  July 2019 Revised  January 2020 Published  June 2020

Fund Project: This research was supported by the German Research Foundation (DFG), Collaborative Research Center SFB-TR 109

We study a Lagrangian numerical scheme for solving a nonlinear drift diffusion equations of the form $ \partial_t u = \partial_x(u \cdot ({\sf c}^*)^\prime[\partial_x \mathit{h}^\prime(u)+ \mathit{v}^\prime]) $, like Fokker-Plank and $ q $-Laplace equations, on an interval. This scheme will consist of a spatio-temporal discretization founded on the formulation of the equation in terms of inverse distribution functions. It is based on the gradient flow structure of the equation with respect to optimal transport distances for a family of costs that are in some sense $ p $-Wasserstein like. Additionally we will show that, under a regularity assumption on the initial data, this also includes a family of discontinuous, flux-limiting cost inducing equations like Rosenau's relativistic heat equation. We show that this discretization inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, a minimum/maximum principle and flux-limitation in the case of the corresponding cost. Convergence in the limit of vanishing mesh size will be proven as the main result. Finally we will present numerical experiments including a numerical convergence analysis.

Citation: Benjamin Söllner, Oliver Junge. A convergent Lagrangian discretization for $ p $-Wasserstein and flux-limited diffusion equations. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4227-4256. doi: 10.3934/cpaa.2020190
References:
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M. Agueh et al., Existence of solutions to degenerate parabolic equations via the MongeKantorovich theory, Adv. Differ. Equ., 10 (2005), 309–360.  Google Scholar

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L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: in Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, 2008.  Google Scholar

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J. D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.  doi: 10.1007/s002110050002.  Google Scholar

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J. CarrilloB. D$\ddot{u}$ringD. Matthes and D. S. McCormick, A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes, J. Sci. Comput., 75 (2018), 1463-1499.  doi: 10.1007/s10915-017-0594-5.  Google Scholar

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

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F. Cavalli and G. Naldi, A wasserstein approach to the numerical solution of the one-dimensional cahn-hilliard equation, Kinet. Relat. Models, 3 (2010), 123-142.  doi: 10.3934/krm.2010.3.123.  Google Scholar

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V. De CiccoN. Fusco and A. Verde, On l1-lower semicontinuity in bv, J. Convex Anal., 12 (2005), 173-185.   Google Scholar

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B. DüringD. 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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E. Giusti and G. H. Williams, Minimal Surfaces and Functions of Bounded Variation, Vol. 2, Springer, 1984.  Google Scholar

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L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM J. Numer. Anal., 43 (2006), 2590-2606.  doi: 10.1137/040608672.  Google Scholar

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M. Leven, Gradientenfluß-basierte Diskretisierung Parabolischer Gleichungen, Ph.D thesis, Inst. für Angew. Math. der Univ., 2002. Google Scholar

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R. MacCamy and E. Socolovsky, A numerical procedure for the porous media equation, Comput. Math. Appl., 11 (1985), 315-219.  doi: 10.1016/0898-1221(85)90156-7.  Google Scholar

[18]

D. Matthes and H. Osberger, Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation, ESAIM Math. Model. Numer. Anal., 48 (2014), 697-726.  doi: 10.1051/m2an/2013126.  Google Scholar

[19]

R. J. McCann and M. Puel, Constructing a relativistic heat flow by transport time steps, Ann. Inst. Henri Poincare (C) Non Linear Anal., 26 (2009), 2539-2580.  doi: 10.1016/j.anihpc.2009.06.006.  Google Scholar

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T. Roessler, Discretizing the Porous Medium Equation Based on Its Gradient Flow Structure: A Consistency Paradox, Universit$\ddot{a}$t Bonn, SFB 611, Singul$\ddot{a}$re Ph$\ddot{a}$nomene und Skalierung in, Technical Report, 2004. Google Scholar

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P. Rosenau, Tempered diffusion: A transport process with propagating fronts and inertial delay, Phys. Rev. A, 46 (1992), R7371. Google Scholar

[22]

R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in banach spaces, Ann. Scuola Norm. Super. Pisa-Cl. Sci. Ser. V, 2 (2003), 395.  Google Scholar

[23]

G. Russo, Deterministic diffusion of particles, Commun. Pure Appl. Math., 43 (1990), 697-733.  doi: 10.1002/cpa.3160430602.  Google Scholar

[24]

F. Santambrogio, Optimal transport for applied mathematicians, Birkäuser, NY, 99–102. doi: 10.1007/978-3-319-20828-2.  Google Scholar

[25]

C. Villani, Topics in Optimal Transportation, Vol. 58, American Mathematical Soc., 2003. doi: 10.1090/gsm/058.  Google Scholar

[26]

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

show all references

References:
[1]

M. Agueh et al., Existence of solutions to degenerate parabolic equations via the MongeKantorovich theory, Adv. Differ. Equ., 10 (2005), 309–360.  Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: in Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, 2008.  Google Scholar

[3]

J. D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.  doi: 10.1007/s002110050002.  Google Scholar

[4]

C. BuddG. CollinsW. Huang and R. Russell, Self–similar numerical solutions of the porous–medium equation using moving mesh methods, Philos. Trans. R. Soc. London Ser. A Math. Phys. Eng., 357 (1999), 1047-1077.  doi: 10.1098/rsta.1999.0364.  Google Scholar

[5]

M. Burger, J. A. Carrillo and M. T. Wolfram et al., A mixed finite element method for nonlinear diffusion equations, Kinet. Relat. Models, 3 (2010). doi: 10.3934/krm.2010.3.59.  Google Scholar

[6]

J. CarrilloB. D$\ddot{u}$ringD. Matthes and D. S. McCormick, A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes, J. Sci. Comput., 75 (2018), 1463-1499.  doi: 10.1007/s10915-017-0594-5.  Google Scholar

[7]

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

[8]

F. Cavalli and G. Naldi, A wasserstein approach to the numerical solution of the one-dimensional cahn-hilliard equation, Kinet. Relat. Models, 3 (2010), 123-142.  doi: 10.3934/krm.2010.3.123.  Google Scholar

[9]

V. De CiccoN. Fusco and A. Verde, On l1-lower semicontinuity in bv, J. Convex Anal., 12 (2005), 173-185.   Google Scholar

[10]

B. DüringD. 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

[11]

L. C. Evans, Partial Differential Equations, Springer, 2013. Google Scholar

[12]

E. Giusti and G. H. Williams, Minimal Surfaces and Functions of Bounded Variation, Vol. 2, Springer, 1984.  Google Scholar

[13]

L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM J. Numer. Anal., 43 (2006), 2590-2606.  doi: 10.1137/040608672.  Google Scholar

[14]

R. JordanD. 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

[15]

D. Kinderlehrer and N. J. Walkington, Approximation of parabolic equations using the wasserstein metric, ESAIM Math. Model. Numer. Anal., 33 (1999), 837-852.  doi: 10.1051/m2an:1999166.  Google Scholar

[16]

M. Leven, Gradientenfluß-basierte Diskretisierung Parabolischer Gleichungen, Ph.D thesis, Inst. für Angew. Math. der Univ., 2002. Google Scholar

[17]

R. MacCamy and E. Socolovsky, A numerical procedure for the porous media equation, Comput. Math. Appl., 11 (1985), 315-219.  doi: 10.1016/0898-1221(85)90156-7.  Google Scholar

[18]

D. Matthes and H. Osberger, Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation, ESAIM Math. Model. Numer. Anal., 48 (2014), 697-726.  doi: 10.1051/m2an/2013126.  Google Scholar

[19]

R. J. McCann and M. Puel, Constructing a relativistic heat flow by transport time steps, Ann. Inst. Henri Poincare (C) Non Linear Anal., 26 (2009), 2539-2580.  doi: 10.1016/j.anihpc.2009.06.006.  Google Scholar

[20]

T. Roessler, Discretizing the Porous Medium Equation Based on Its Gradient Flow Structure: A Consistency Paradox, Universit$\ddot{a}$t Bonn, SFB 611, Singul$\ddot{a}$re Ph$\ddot{a}$nomene und Skalierung in, Technical Report, 2004. Google Scholar

[21]

P. Rosenau, Tempered diffusion: A transport process with propagating fronts and inertial delay, Phys. Rev. A, 46 (1992), R7371. Google Scholar

[22]

R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in banach spaces, Ann. Scuola Norm. Super. Pisa-Cl. Sci. Ser. V, 2 (2003), 395.  Google Scholar

[23]

G. Russo, Deterministic diffusion of particles, Commun. Pure Appl. Math., 43 (1990), 697-733.  doi: 10.1002/cpa.3160430602.  Google Scholar

[24]

F. Santambrogio, Optimal transport for applied mathematicians, Birkäuser, NY, 99–102. doi: 10.1007/978-3-319-20828-2.  Google Scholar

[25]

C. Villani, Topics in Optimal Transportation, Vol. 58, American Mathematical Soc., 2003. doi: 10.1090/gsm/058.  Google Scholar

[26]

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

Figure 1.  Experiment: p-Wasserstein cost, linear diffusion. Left: Approximate densities $ u(t, \cdot) $ at $ t = 0.01\cdot 10^k $, $ k = 0, 0.12, 0.24, \ldots, \log_{10}(200) $, initial mass uniformly distributed on $ [-0.3, 0.3] $. Right: the corresponding characteristics
Figure 2.  Experiment: p-Wasserstein cost, linear diffusion. Left: Approximate densities $ u(t, \cdot) $ at $ t = 0.01\cdot 10^k $, $ k = 0, 0.12, 0.24, \ldots, \log_{10}(200) $, initial mass uniformly distributed on $ [-3, -2.4] $. Right: the corresponding characteristics
Figure 3.  Experiment: relativistic cost, linear diffusion. Left: Approximate densities $ u(t, \cdot) $ for $ t = 0.01\cdot 10^k $, $ k = 0, 0.12, 0.24, \ldots, \log_{10}(200) $, initial mass uniformly distributed on $ [-0.3, 0.3] $. Right: the corresponding characteristics (dashed: speed of light)
Figure 4.  Convergence analysis: relativistic cost, linear diffusion. $ L^1 $-error of the inverse distribution function in dependence of the grid size (left), and in dependence of the time step (right)
Figure 5.  Experiment: $ q $-Laplace ($ p = \frac43, m = \frac53 $). Left: Approximate densities $ u(t,\cdot) $ for $ t = 0.01\cdot 10^k $, $ k = 0,0.12,0.24,\ldots,\log_{10}(200) $, initial mass uniformly distributed on $ [-0.3,0.3] $. Right: the corresponding characteristics(dashed: speed of light)
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