A second-order accurate numerical scheme for a time-fractional Fokker–Planck equation

• Published in: IMA journal of numerical analysis Vol. 43; no. 4; pp. 2115 - 2136
• Main Authors: Mustapha, Kassem, Knio, Omar M, Le Maître, Olivier P
• Format: Journal Article
• Language: English
• Published: Oxford University Press 03-08-2023; Oxford University Press (OUP)
• Subjects: Analysis of PDEs; Mathematics; Numerical Analysis; time discretizations; graded meshes; fractional Fokker–Planck; stability and error analysis; finite elements
• ISSN: 0272-4979; 1464-3642
• DOI: 10.1093/imanum/drac031

Abstract
A second-order accurate time-stepping scheme for solving a time-fractional Fokker–Planck equation of order $\alpha \in (0, 1)$, with a general driving force, is investigated. A stability bound for the semidiscrete solution is obtained for $\alpha \in (1/2,1)$ via a novel and concise approach. Our stability estimate is $\alpha $-robust in the sense that it remains valid in the limiting case where $\alpha $ approaches $1$ (when the model reduces to the classical Fokker–Planck equation), a limit that presents practical importance. Concerning the error analysis, we obtain an optimal second-order accurate estimate for $\alpha \in (1/2,1)$. A time-graded mesh is used to compensate for the singular behavior of the continuous solution near the origin. The time-stepping scheme scheme is associated with a standard spatial Galerkin finite element discretization to numerically support our theoretical contributions. We employ the resulting fully discrete computable numerical scheme to perform some numerical tests. These tests suggest that the imposed time-graded meshes assumption could be further relaxed, and we observe second-order accuracy even for the case $\alpha \in (0,1/2]$, that is, outside the range covered by the theory.