Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient-flow structure. (English) Zbl 1467.35317
Summary: We propose fully discrete, implicit-in-time finite-volume schemes for a general family of non-linear and non-local Fokker-Planck equations with a gradient-flow structure, usually known as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivity-preservation and energy-dissipation properties, essential for their practical use. The first-order scheme verifies these properties unconditionally for general non-linear diffusions and interaction potentials, while the second-order scheme does so provided a CFL condition holds. Sweeping dimensional splitting permits the efficient construction of these schemes in higher dimensions while preserving their structural properties. Numerical experiments validate the schemes and show their ability to handle complicated phenomena typical in aggregation-diffusion equations, such as free boundaries, metastability, merging and phase transitions.
MSC:
35Q84 | Fokker-Planck equations |
35B09 | Positive solutions to PDEs |
35R09 | Integro-partial differential equations |
45K05 | Integro-partial differential equations |
65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
82C21 | Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics |
82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |