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

Abstract

A Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planck equations in space dimensions $d\ge 2$ 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.

32 pages, 7 figures

References in corpus (5)

• Numerical Study of a Particle Method for Gradient Flows
• Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
• Convergence of a particle method for diffusive gradient flows in one dimension
• A fully discrete variational scheme for solving nonlinear Fokker-Planck equations in higher space dimensions
• Convergence of a fully discrete variational scheme for a thin-film equation

Cited by in corpus (12)

• A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance
• Field Theory of Reaction-Diffusion: Mass Action with an Energetic Variational Approach
• On Lagrangian schemes for porous medium type generalized diffusion equations: a discrete energetic variational approach
• Primal dual methods for Wasserstein gradient flows
• On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobility
• High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems
• Particle-based Energetic Variational Inference
• A dynamic mass transport method for Poisson-Nernst-Planck equations
• Lagrangian schemes for Wasserstein gradient flows
• Numerical study of Bose-Einstein condensation in the Kaniadakis-Quarati model for bosons
• A Variational Lagrangian Scheme for a Phase Field Model: A Discrete Energetic Variational Approach
• A convergent Lagrangian discretization for $p$-Wasserstein and flux-limited diffusion equations