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07-11-2017 | Issue 3/2018 Open Access

Journal of Scientific Computing 3/2018

A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes

Journal:
Journal of Scientific Computing > Issue 3/2018
Authors:
José A. Carrillo, Bertram Düring, Daniel Matthes, David S. McCormick
Important notes
JAC acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) under Grant No. EP/P031587/1, by the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award and by the National Science Foundation (NSF) under Grant No. RNMS11-07444 (KI-Net). DM was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”. BD and DSMcC were supported by the Leverhulme Trust research project grant “Novel discretisations for higher-order nonlinear PDE” (RPG-2015-69).

Abstract

A Lagrangian numerical scheme for solving nonlinear degenerate Fokker–Planck equations in space dimensions \(d\ge 2\) is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies and given external potentials, e.g. the porous medium equation and the fast diffusion equation. The key ingredient in our approach is the gradient flow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient flow, and these lead to weak compactness of the trajectories in the continuous limit. Consistency is analyzed in the planar situation, \(d=2\). A variety of numerical experiments for the porous medium equation indicates that the scheme is well-adapted to track the growth of the solution’s support.

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