A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes

Carrillo, José A, Düring, Bertram, Matthes, Daniel and McCormick, David S (2018) A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes. Journal of Scientific Computing, 75 (3). pp. 1463-1499. ISSN 0885-7474

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Abstract

A Lagrangian numerical scheme for solving nonlinear degenerate Fokker{Planck equations in space dimensions d>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 ow 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 ow, 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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics
Depositing User: Billy Wichaidit
Date Deposited: 27 Oct 2017 14:34
Last Modified: 04 May 2018 16:23
URI: http://sro.sussex.ac.uk/id/eprint/70715

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Project NameSussex Project NumberFunderFunder Ref
Novel discretisations of higher-order nonlinear PDEG1603LEVERHULME TRUSTRPG-2015-069