TY - JOUR
T1 - A second-order accurate numerical scheme for a time-fractional Fokker-Planck equation
AU - Mustapha, Kassem
AU - Knio, Omar M.
AU - Le Maître, Olivier P.
N1 - Publisher Copyright:
© 2022 The Author(s). Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
PY - 2023/7/1
Y1 - 2023/7/1
N2 - A second-order accurate time-stepping 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 semidiscrete solution is obtained for α ∈ (1/2, 1) via a novel and concise approach. Our stability estimate is -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 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 α ∈ (1/2, 1), that is, outside the range covered by the theory.
AB - A second-order accurate time-stepping 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 semidiscrete solution is obtained for α ∈ (1/2, 1) via a novel and concise approach. Our stability estimate is -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 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 α ∈ (1/2, 1), that is, outside the range covered by the theory.
KW - finite elements
KW - fractional Fokker-Planck
KW - graded meshes
KW - stability and error analysis
KW - time discretizations
UR - http://www.scopus.com/inward/record.url?scp=85168306969&partnerID=8YFLogxK
U2 - 10.1093/imanum/drac031
DO - 10.1093/imanum/drac031
M3 - Article
AN - SCOPUS:85168306969
SN - 0272-4979
VL - 43
SP - 2115
EP - 2136
JO - IMA Journal of Numerical Analysis
JF - IMA Journal of Numerical Analysis
IS - 4
ER -