A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes
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...
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
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Springer Nature
2017
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| _version_ | 1797058319713566720 |
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| author | Carrillo de la Plata, JA During, B Matthes, D McCormick, DS |
| author_facet | Carrillo de la Plata, JA During, B Matthes, D McCormick, DS |
| author_sort | Carrillo de la Plata, JA |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T19:48:48Z |
| format | Journal article |
| id | oxford-uuid:23413e17-6f3e-4a63-9fed-b865797c0d3d |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T19:48:48Z |
| publishDate | 2017 |
| publisher | Springer Nature |
| record_format | dspace |
| spelling | oxford-uuid:23413e17-6f3e-4a63-9fed-b865797c0d3d2022-03-26T11:43:22ZA Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex MeshesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:23413e17-6f3e-4a63-9fed-b865797c0d3dEnglishSymplectic ElementsSpringer Nature2017Carrillo de la Plata, JADuring, BMatthes, DMcCormick, DSA 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. |
| spellingShingle | Carrillo de la Plata, JA During, B Matthes, D McCormick, DS A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes |
| title | A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes |
| title_full | A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes |
| title_fullStr | A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes |
| title_full_unstemmed | A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes |
| title_short | A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes |
| title_sort | lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes |
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