Higher order finite difference method for the reaction and anomalous-diffusion equation☆☆☆
Higher order finite difference method for the reaction and anomalous-diffusion equation☆☆☆
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DOI：
10.1016/j.apm.2013.12.002
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发表时间：
2014-08
影响因子：
5
通讯作者：
Changpin Li;Heng-fei Ding
中科院分区：
文献类型：
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作者：
Changpin Li;Heng-fei Ding
本文主要研究反应反常扩散方程的高阶有限差分方法。根据Riemann-Liouville导数与Grünwald-Letnikov导数在适当的光滑条件下的等价性，导出了Riemann-Liouville分数阶导数的二阶差分近似。对空间二阶导数采用四阶紧致差分逼近。利用傅立叶方法分析了该格式的可解性、条件稳定性和收敛性。得到了收敛阶为O(τ2+h4)，其中τ为时间步长，h为空间步长。最后进行了数值实验，结果表明数值结果与理论分析吻合较好。
In this paper, our aim is to study the high order finite difference method for the reaction and anomalous-diffusion equation. According to the equivalence of the Riemann–Liouville and Grünwald–Letnikov derivatives under the suitable smooth condition, a second-order difference approximation for the Riemann–Liouville fractional derivative is derived. A fourth-order compact difference approximation for second-order derivative in spatial is used. We analyze the solvability, conditional stability and convergence of the proposed scheme by using the Fourier method. Then we obtain that the convergence order is O (τ 2+ h 4), where τ is the temporal step length and h is the spatial step length. Finally, numerical experiments are presented to show that the numerical results are in good agreement with the theoretical analysis.