A non-standard finite difference method for space fractional advection–diffusion equation
A non-standard finite difference method for space fractional advection–diffusion equation
复制标题
空间分数平流扩散方程的非标准有限差分法
DOI：
10.1002/num.22734
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Wang Qi
中科院分区：
文献类型：
--
作者：
Wang Qi
本文提出了求解空间分数阶平流扩散方程的非标准有限差分格式。利用Fourier-Von Neumann方法，证明了非标准有限差分格式是无条件稳定的。进一步讨论了数值方法的收敛性，并给出了收敛阶。数值算例表明，与经典数值方法相比，非标准有限差分法能有效地减小最大误差，提高数值解的精度。此外，我们发现我们的数值方案非常灵活，当我们同时优化时间和空间的分母函数时，性能是最好的。这些研究表明，非标准有限差分格式对于求解分数阶偏微分方程是可行和有效的。
In this paper, a non‐standard finite difference scheme is developed to solve the space fractional advection–diffusion equation. By using Fourier–Von Neumann method, we prove that non‐standard finite difference scheme is unconditionally stable. We further discuss the convergence of numerical method and give the order of convergence. The numerical examples show that the non‐standard finite difference method can effectively reduce the maximum error and improve the accuracy of numerical solution in contrast to classical numerical methods. Moreover, we find that our numerical scheme is very flexible, when we optimize the denominator function of time and space simultaneously, the performance is the best. These studies show that the non‐standard finite difference scheme is feasible and efficient for solving fractional partial differential equations.