@Article{CiCP-16-516,
author = {Minghua Chen and Weihua Deng},
title = {Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators},
journal = {Communications in Computational Physics},
year = {2014},
volume = {16},
number = {2},
pages = {516--540},
abstract = {<p style="text-align: justify;">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 as first order schemes but the accuracy is greatly improved. Using
the fractional linear multistep methods, Lubich obtains the ν-th order (ν≤6) approximations
of the α-th derivative (α&gt;0) or integral (α&lt;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 α∈(1,2) for time dependent problem. By
weighting and shifting Lubich&#39;s 2nd order discretization scheme, in [Chen &amp; 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&#39;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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.120713.280214a},
url = {http://global-sci.org/intro/article_detail/cicp/7052.html}
}