FOURTH ORDER ACCURATE SCHEME FOR THE SPACE FRACTIONAL DIFFUSION EQUATIONS

• Published in: SIAM journal on numerical analysis Vol. 52; no. 3; pp. 1418 - 1438
• Main Authors: CHEN, MINGHUA, DENG, WEIHUA
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2014
• Subjects: Accuracy; Algebra; Applied mathematics; Approximation; Boundary conditions; Coefficients; Convergence; Derivatives; Diffusion; Eigenvalues; Generating function; Integers; Mathematical analysis; Mathematical models; Numerical schemes; Operators; Time dependence; Truncation errors; Variable coefficients
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/130933447

Because of the nonlocal properties of fractional operators, higher order schemes play a more important role in discretizing fractional derivatives than classical ones. The striking feature is that higher order schemes of fractional derivatives can keep the same computation cost with first order schemes but greatly improve the accuracy. Nowadays, there are already two types of second order discretization schemes for space fractional derivatives: the first type is given and discussed in [Sousa and Li, arXiv:1109.2345v1, 2011; Chen and Deng, Appl. Math. Model., 38 (2014), pp. 3244– 3259; Chen, Deng, and Wu, Appl. Numer. Math., 70 (2013), pp. 22–41]; and the second type is a class of schemes presented in [Tian, Zhou, and Deng, Math. Comp., to appear; also available online from arXiv:1201.5949, 2012]. The core object of this paper is to derive a class of fourth order approximations, called the weighted and shifted Lubich difference operators, for space fractional derivatives. Then we use the derived schemes to solve the space fractional diffusion equation with variable co-efficients in one-dimensional and two-dimensional cases. And the unconditional stability and the convergence with the global truncation error 𝓞(τ2 + h4) are theoretically proved and numerically verified.