Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T22:11:04.019Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of a variational Lagrangian scheme for a nonlineardrift diffusion equation∗

Published online by Cambridge University Press:  07 February 2014

Daniel Matthes  and
Horst Osberger
Daniel Matthes
Affiliation:
Zentrum Mathematik, TU München, Boltzmannstr. 3, 85748 Garching, Germany.. matthes@ma.tum.de; osberger@ma.tum.de
Horst Osberger
Affiliation:
Zentrum Mathematik, TU München, Boltzmannstr. 3, 85748 Garching, Germany.. matthes@ma.tum.de; osberger@ma.tum.de
Get access

Abstract

We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusionequation on an interval. The discretization is based on the equation’s gradient flowstructure with respect to the Wasserstein distance. The scheme inherits various propertiesfrom the continuous flow, like entropy monotonicity, mass preservation, metric contractionand minimum/ maximum principles. As the main result, we give a proof of convergence in thelimit of vanishing mesh size under a CFL-type condition. We also present results fromnumerical experiments.

Keywords

Lagrangian discretizationnonlinear Fokker-Planck equationgradient flowWasserstein metric
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 697 - 726
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
Ambrosio, L., Lisini, S. and Savaré, G., Stability of flows associated to gradient vector fields and convergence of iterated transport maps. Manuscripta Math. 121 (2006) 1–50. Google Scholar
Benamou, J.-D. and Brenier, Y., A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375–393. Google Scholar
Blanchet, A., Calvez, V. and Carrillo, J.A., Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model. SIAM J. Numer. Anal. 46 (2008) 691–721. Google Scholar
Budd, C.J., Collins, G.J., Huang, W.Z. and Russell, R.D., Self-similar numerical solutions of the porous-medium equation using moving mesh methods. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357 (1999) 1047–1077. Google Scholar
Burger, M., Carrillo, J.A. and Wolfram, M.-T., A mixed finite element method for nonlinear diffusion equations. Kinet. Relat. Models 3 (2010) 59–83. Google Scholar
Carrillo, J.A. and Moll, J.S., Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms. SIAM J. Sci. Comput. 31 (2009/2010) 4305–4329. Google Scholar
Cavalli, F. and Naldi, G., A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation. Kinet. Relat. Models 3 (2010) 123–142. Google Scholar
Düring, B., Matthes, D. and Milišić, J.P., A gradient flow scheme for nonlinear fourth order equations. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 935–959. Google Scholar
Evans, L.C., Savin, O. and Gangbo, W., Diffeomorphisms and nonlinear heat flows. SIAM J. Math. Anal. 37 (2005) 737–751. Google Scholar
E. Giusti, Minimal surfaces and functions of bounded variation, vol. 80, Monographs in Mathematics. Birkhäuser Verlag, Basel (1984).
Gosse, L. and Toscani, G., Identification of asymptotic decay to self-similarity for one-dimensional filtration equations. SIAM J. Numer. Anal. 43 (2006) 2590–2606 (electronic). Google Scholar
Jordan, R., Kinderlehrer, D. and Otto, F., The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1–17. Google Scholar
Kinderlehrer, D. and Walkington, N.J., Approximation of parabolic equations using the Wasserstein metric. ESAIM: M2AN 33 (1999) 837–852. Google Scholar
M. Leven, Gradientenfluß-basierte diskretisierung parabolischer gleichungen, diplomarbeit, Universität Bonn (2002).
MacCamy, R.C. and Socolovsky, E., A numerical procedure for the porous media equation. Hyperbolic partial differential equations, II. Comput. Math. Appl. 11 (1985) 315–319. Google Scholar
McCann, R.J., A convexity principle for interacting gases. Adv. Math. 128 (1997) 153–179. Google Scholar
Otto, F., The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26 (2001) 101–174. Google Scholar
T. Roessler, Discretizing the porous medium equation based on its gradient flow structure – a consistency paradox, Technical report 150, Sonderforschungsbereich 611, May 2004.Available online at http://sfb611.iam.uni-bonn.de/uploads/150-komplett.pdf.
Rossi, R. and Savaré, G., Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 (2003) 395–431. Google Scholar
Russo, G., Deterministic diffusion of particles. Commun. Pure Appl. Math. 43 (1990) 697–733. Google Scholar
Serfaty, S., Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. 31 (2011) 1427-1451. Google Scholar
C. Villani, Topics in optimal transportation, in vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2003).
Westdickenberg, M. and Wilkening, J., Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations. ESAIM: M2AN 44 (2010) 133–166. Google Scholar