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Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations: A Second-Order Scheme

Part of: Miscellaneous topics - Partial differential equations General theory in ordinary differential equations Partial differential equations, initial value and time-dependent initial-boundary value problems Real functions Computational aspects Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  28 July 2017

Yonggui Yan ,
Zhi-Zhong Sun  and
Jiwei Zhang
Yonggui Yan*
Affiliation:
Beijing Computational Science Research Center, Beijing 100193, China
Zhi-Zhong Sun*
Affiliation:
Department of Mathematics, Southeast University, Nanjing 211189, China
Jiwei Zhang*
Affiliation:
Beijing Computational Science Research Center, Beijing 100193, China
*
*Corresponding author. Email addresses:yan_yonggui@csrc.ac.cn (Y. Yan), zzsun@seu.edu.cn (Z.-Z. Sun), jwzhang@csrc.ac.cn (J. Zhang)
*Corresponding author. Email addresses:yan_yonggui@csrc.ac.cn (Y. Yan), zzsun@seu.edu.cn (Z.-Z. Sun), jwzhang@csrc.ac.cn (J. Zhang)
*Corresponding author. Email addresses:yan_yonggui@csrc.ac.cn (Y. Yan), zzsun@seu.edu.cn (Z.-Z. Sun), jwzhang@csrc.ac.cn (J. Zhang)
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Abstract

The fractional derivatives include nonlocal information and thus their calculation requires huge storage and computational cost for long time simulations. We present an efficient and high-order accurate numerical formula to speed up the evaluation of the Caputo fractional derivative based on the L2-1σ formula proposed in [A. Alikhanov, J. Comput. Phys., 280 (2015), pp. 424-438], and employing the sum-of-exponentials approximation to the kernel function appeared in the Caputo fractional derivative. Both theoretically and numerically, we prove that while applied to solving time fractional diffusion equations, our scheme not only has unconditional stability and high accuracy but also reduces the storage and computational cost.

Keywords

Caputo fractional derivativefractional diffusion equationstability analysissum-of-exponentials approximationfast algorithm

MSC classification

Secondary: 11T23: Exponential sums 26A33: Fractional derivatives and integrals 33F05: Numerical approximation and evaluation 34A08: Fractional differential equations 35R11: Fractional partial differential equations 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 4 , October 2017 , pp. 1028 - 1048
Copyright
Copyright © Global-Science Press 2017 

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