Work

Numerical methods and regularity properties for viscosity solutions of nonlocal in space and time diffusion equations

Year: 2025

Type: article

Abstract: We consider a general family of nonlocal in space and time diffusion equations with space-time dependent diffusivity and prove convergence of finite difference schemes in the context of viscosity solutions under very mild conditions. The proofs, based on regularity properties and compactness arguments on the numerical solution, allow to inherit a number of interesting results for the limit equation. More precisely, assuming Hölder regularity only on the initial condition, we prove convergence of the scheme, space-time Hölder regularity of the solution, depending on the fractional orders of the operators, as well as specific blow up rates of the first time derivative. The schemes’ performance is further numerically verified using both constructed exact solutions and realistic examples. Our experiments show that multithreaded implementation yields an efficient method to solve nonlocal equations numerically.

Source: IMA Journal of Numerical Analysis

DOI: 10.1093/imanum/draf011

Authors: Félix del Teso, Łukasz Płociniczak

Institutions: Universidad Autónoma de Madrid, Wrocław University of Science and Technology

Language: English

Topic: Fractional Differential Equations Solutions

Subfield: Modeling and Simulation

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Vetenskapsrådet, Narodowym Centrum Nauki, Agencia Estatal de Investigación

Awards: Institut Mittag-Leffler, G4233400527