Commun. Comput. Phys., 16 (2014), pp. 516-540.


Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators

Minghua Chen 1, Weihua Deng 1*

1 School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P.R. China.

Received 12 July 2013; Accepted (in revised version) 28 February 2014
Available online 12 June 2014
doi:10.4208/cicp.120713.280214a

Abstract

High order discretization schemes play more important role in fractional operators than classical ones. This is because usually for classical derivatives the stencil for high order discretization schemes is wider than low order ones; but for fractional operators the stencils for high order schemes and low order ones are the same. Then using high order schemes to solve fractional equations leads to almost the same computational cost with first order schemes but the accuracy is greatly improved. Using the fractional linear multistep methods, Lubich obtains the $\nu$-th order ($\nu\leq 6$) approximations of the $\alpha$-th derivative ($\alpha>0$) or integral ($\alpha<0$) [Lubich, SIAM J. Math. Anal., 17, 704-719, 1986], because of the stability issue the obtained scheme can not be directly applied to the space fractional operator with $\alpha \in(1,2)$ for time dependent problem. By weighting and shifting Lubich's 2nd order discretization scheme, in [Chen \& Deng, SINUM, arXiv:1304.7425] we derive a series of effective high order discretizations for space fractional derivative, called WSLD operators there. As the sequel of the previous work, we further provide new high order schemes for space fractional derivatives by weighting and shifting Lubich's 3rd and 4th order discretizations. In particular, we prove that the obtained 4th order approximations are effective for space fractional derivatives. And the corresponding schemes are used to solve the space fractional diffusion equation with variable coefficients.

AMS subject classifications: 26A33, 03F35, 47B37, 65M12

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Key words: Fractional derivatives, high order scheme, weighted and shifted Lubich difference (WSLD) operators, numerical stability.

*Corresponding author.
Email: dengwh@lzu.edu.cn (W. Deng)
 

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