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

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Carrillo, Jose A., Düring, Bertram, Matthes, Daniel and McCormick, David S. (2017) A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes. Journal of Scientific Computing, 75 (3). pp. 1463-1499. doi:10.1007/s10915-017-0594-5

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Official URL: https://doi.org/10.1007/s10915-017-0594-5

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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 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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Lagrange spectrum, Mathematical optimization, Nonlinear partial differential operators , Fokker-Planck equation
Journal or Publication Title: Journal of Scientific Computing
Publisher: Springer Link
Official Date: 7 November 2017
Dates:
DateEvent
7 November 2017Published
27 October 2017Accepted
Date of first compliant deposit: 26 October 2020
Volume: 75
Number: 3
Page Range: pp. 1463-1499
DOI: 10.1007/s10915-017-0594-5
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
EP/P031587/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
UNSPECIFIEDRoyal Societyhttp://dx.doi.org/10.13039/501100000288
UNSPECIFIEDWolfson Foundationhttp://dx.doi.org/10.13039/501100001320
RNMS11-07444 (KI-Net)National Science Foundationhttp://dx.doi.org/10.13039/501100008982
TRR 109[DFG] Deutsche Forschungsgemeinschafthttp://dx.doi.org/10.13039/501100001659
RPG-2015-69Leverhulme Trusthttp://dx.doi.org/10.13039/501100000275
Open Access Version:
  • https://doi.org/10.1007/s10915-017-0594-...

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