Journal
JOURNAL OF SCIENTIFIC COMPUTING
Volume 76, Issue 1, Pages 583-609Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10915-017-0631-4
Keywords
Fractional differential equation; Caputo derivative; Weak singularity; Fitted scheme; Graded mesh
Categories
Funding
- Institute of Mathematics and Applications (IUMA) [MTM2016-75139-R]
- Diputacion General de Aragon
- National Natural Science Foundation of China [91430216, NSAF-U1530401]
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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.
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