WSLD operators II: the new fourth order difference approximations for space Riemann-Liouville derivative

TL;DR

This paper develops new fourth order difference schemes for space Riemann-Liouville derivatives, improving accuracy while maintaining computational efficiency, and applies them to space fractional diffusion equations with variable coefficients.

Contribution

It introduces novel fourth order discretization schemes for space fractional derivatives by extending previous second and third order methods, demonstrating their effectiveness in solving fractional diffusion equations.

Findings

The 4th order schemes are proven effective for space fractional derivatives.

The schemes improve accuracy without increasing computational cost significantly.

Application to variable coefficient fractional diffusion equations shows practical utility.

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, arXiv:1304.7425] we derive a series of effective high order discretizations for space fractional derivative, called WSLD opeartors 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.