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

A second-order accurate time-stepping scheme for solving a time-fractional Fokker-Planck equation of order alpha is an element of (0, 1), with a general driving force, is investigated. A stability bound for the semidiscrete solution is obtained for alpha is an element of(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 is an element of (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 is an element of (0,1/2], that is, outside the range covered by the theory.

Automated summary

This study investigates a second-order accurate time-stepping scheme for solving a time-fractional Fokker-Planck equation of order alpha is an element of (0, 1) with a general driving force. A stability bound for the semidiscrete solution is obtained for alpha is an element of (1/2,1) using a novel and concise approach. The study also obtains an optimal second-order accurate estimate for alpha is an element of (1/2,1) and uses a time-graded mesh to compensate for the singular behavior of the continuous solution near the origin. Numerical tests suggest that the time-graded mesh assumption can be further relaxed.