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

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Title: A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes
Authors: Carrillo de la Plata, J
Düring, B
Matthes, D
McCormick, DS
Item Type: Journal Article
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 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.
Issue Date: 1-Jun-2018
Date of Acceptance: 25-Oct-2017
URI: http://hdl.handle.net/10044/1/52560
DOI: https://dx.doi.org/10.1007/s10915-017-0594-5
ISSN: 0885-7474
Publisher: Springer Verlag
Start Page: 1463
End Page: 1499
Journal / Book Title: Journal of Scientific Computing
Volume: 75
Issue: 3
Copyright Statement: © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Sponsor/Funder: The Royal Society
Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: WM120001
EP/P031587/1
Keywords: 0102 Applied Mathematics
0103 Numerical And Computational Mathematics
0802 Computation Theory And Mathematics
Applied Mathematics
Publication Status: Published
Online Publication Date: 2017-11-07
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences



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