A second-order accurate numerical scheme for a time-fractional Fokker-Planck equation
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A time-stepping L1 scheme for solving a time fractional Fokker-Planck equation of order α∈ (0, 1), with a general driving force, is investigated. A stability bound for the semi-discrete solution is obtained for α∈(1/2,1) via a novel and concise approach. Our stability estimate is α-robust in the sense that it remains valid in the limiting case where α 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 α∈(1/2,1). A time-graded mesh is used to compensate for the singular behavior of the continuous solution near the origin. The L1 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 α∈(0,1/2], that is, outside the range covered by the theory.
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