WSLD算子II：空间Riemann-Liouville导数的新型四阶差分近似
WSLD operators II: the new fourth order difference approximations for space Riemann-Liouville derivative
AI总结：
本文基于Lubich的线性多步方法，通过加权和位移得到新的高阶差分方案，用于空间分数导数的近似，并证明四阶近似在空间分数扩散方程中的有效性。
摘要：
高阶离散化方案在分数算子中比经典算子更具重要性，因为通常经典导数的高阶离散化方案的 stencil 宽于低阶方案；但分数算子的高阶和低阶方案的 stencil 是相同的。使用高阶方案求解分数方程几乎与一阶方案有相同的计算成本，但精度显著提高。通过加权和位移Lubich的2阶离散化方案，在[Chen & Deng, arXiv:1304.7425]中推导出一系列有效的高阶离散化方案，称为WSLD算子。作为前一工作的延续，本文进一步通过加权和位移Lubich的3阶和4阶离散化方案，提供新的高阶方案用于空间分数导数。特别地，证明得到的四阶近似在空间分数导数中是有效的。相应的方案用于求解具有变量系数的空间分数扩散方程。
英文摘要：
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 $ν$-th order ($ν\leq 6$) approximations of the $α$-th derivative ($α>0$) or integral ($α<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 $α\in(1,2)$ 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.
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