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

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journal contribution
posted on 2023-06-13, 15:16 authored by José A Carrillo, Bertram Duering, Daniel Matthes, David S McCormick
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.

Funding

Novel discretisations of higher-order nonlinear PDE; G1603; LEVERHULME TRUST; RPG-2015-069

History

Publication status

  • Published

File Version

  • Published version

Journal

Journal of Scientific Computing

ISSN

0885-7474

Publisher

Springer Verlag

Issue

3

Volume

75

Page range

1463-1499

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Numerical Analysis and Scientific Computing Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2017-10-27

First Open Access (FOA) Date

2018-05-04

First Compliant Deposit (FCD) Date

2017-10-27

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