A Fitted Scheme for a Caputo Initial-Boundary Value Problem

Autores

Gracia, J; O'Riordan, E; Stynes, M

Revista

JOURNAL OF SCIENTIFIC COMPUTING

Año: 2018 Volumen: 76 (1) Páginas: 583-609

Editor:

SPRINGER/PLENUM PUBLISHERS

DOI:

10.1007/s10915-017-0631-4

Resumen

In this paper we consider an initial-boundary value problem with a Caputo time derivative of order . The solution typically exhibits a weak singularity near the initial time and this causes a reduction in the orders of convergence of standard schemes. To deal with this singularity, the solution is computed with a fitted difference scheme on a graded mesh. The convergence of this scheme is analysed using a discrete maximum principle and carefully chosen barrier functions. Sharp error estimates are proved, which show an enhancement in the convergence rate compared with the standard L1 approximation on uniform meshes, and also indicate an optimal choice for the mesh grading. This optimal mesh grading is less severe than the optimal grading for the standard L1 scheme. Furthermore, the dependence of the error on the final time forms part of our error estimate. Numerical experiments are presented which corroborate our theoretical results.

Palabras clave

Fractional ; differential ; equation ; Caputo ; derivative ; Weak ; singularity ; Fitted ; scheme ; Graded ; mesh

Afiliación

Stynes, M (Reprint Author), Beijing Computat Sci Res Ctr, Appl & Computat Math Div, Beijing 100193, Peoples R China.
Gracia, J. L., Univ Zaragoza, Dept Appl Math, Torres Quevedo Bldg,Campus Rio Ebro, Zaragoza 50018, Spain.
O'Riordan, E., Dublin City Univ, Sch Math Sci, Dublin 9, Ireland.
Stynes, M., Beijing Computat Sci Res Ctr, Appl & Computat Math Div, Beijing 100193, Peoples R China.