WSGD-OSC Scheme for Two-Dimensional Distributed Order Fractional Reaction-Diffusion Equation
WSGD-OSC Scheme for Two-Dimensional Distributed Order Fractional Reaction-Diffusion Equation
复制标题
二维分布阶分数反应扩散方程的 WSGD-OSC 方案
DOI：
10.1007/s10915-018-0672-3
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发表时间：
2018
影响因子：
2.5
通讯作者：
Xu Da
中科院分区：
文献类型：
--
作者：
Yang Xuehua;Zhang Haixiang;Xu Da
本文讨论了二维分布阶时间分数阶反应扩散方程的一种新的数值近似。结合时间上的加权移位grasswald差分（WSGD）近似思想（Tian et al. in Math compuput 84:1703-1727, 2015; Wang and Vong in J compuput Phys 277:1-15, 2014），在空间上建立正交样条配置（OSC）方法。详细分析表明，所提方案具有无条件稳定性和收敛性，其收敛阶分别与时间步长、分布阶变量步长、空间步长和空间多项式度有关。有趣的是，我们证明了所提出的WSGD-OSC方案在时间上收敛于二阶，其中先前提出的OSC方案（Fairweather et al. in J Sci Comput 65:1217-1239, 2015; Yang et al. in J Comput Phys 256:824-837, 2014）最多可以实现阶的时间精度，阶依赖于方程中分数阶导数的阶数，通常小于2。一些数值结果也证实了我们的理论预测。
In this paper, a new numerical approximation is discussed for the two-dimensional distributed-order time fractional reaction–diffusion equation. Combining with the idea of weighted and shifted Grünwald difference (WSGD) approximation (Tian et al. in Math Comput 84:1703–1727, 2015; Wang and Vong in J Comput Phys 277:1–15, 2014) in time, we establish orthogonal spline collocation (OSC) method in space. A detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order, whereandrare, respectively the time step size, step size in distributed-order variable, space step size, and polynomial degree of space. Interestingly, we prove that the proposed WSGD-OSC scheme converges with the second-order in time, where OSC schemes proposed previously (Fairweather et al. in J Sci Comput 65:1217–1239, 2015; Yang et al. in J Comput Phys 256:824–837, 2014) can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two. Some numerical results are also given to confirm our theoretical prediction.