Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems
Abstract
In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation.
Cite
@article{arxiv.2105.11146,
title = {Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems},
author = {Daniel Adams and Manh Hong Duong and Goncalo dos Reis},
journal= {arXiv preprint arXiv:2105.11146},
year = {2022}
}
Comments
26 pages. significant revision from the previous versions