A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes
This work provides a novel numerical method for a class of nonlinear diffusion equations, enabling accurate simulation of moving boundaries in higher dimensions.
The paper presents a Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planck equations in dimensions d≥2, using moving simplex meshes. The scheme inherits energy estimates from the gradient flow structure and demonstrates accurate tracking of solution support growth for the porous medium equation in numerical experiments.
A Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planck equations in space dimensions $d\ge2$ 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.