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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
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 | |||||||||||||||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > 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: |
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| 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 | |||||||||||||||||||||
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