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A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes
Version 2 2023-06-13, 15:16
Version 1 2023-06-09, 08:29
journal contribution
posted on 2023-06-13, 15:16 authored by José A Carrillo, Bertram Duering, Daniel Matthes, David S McCormickA 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 ComputingISSN
0885-7474Publisher
Springer VerlagExternal DOI
Issue
3Volume
75Page range
1463-1499Department 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
2017-10-27First Open Access (FOA) Date
2018-05-04First Compliant Deposit (FCD) Date
2017-10-27Usage metrics
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