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A non‐standard finite difference method for space fractional advection–diffusion equation
Numerical Methods for Partial Differential Equations ( IF 3.9 ) Pub Date : 2020-12-23 , DOI: 10.1002/num.22734 Ziting Liu 1 , Qi Wang 1
Numerical Methods for Partial Differential Equations ( IF 3.9 ) Pub Date : 2020-12-23 , DOI: 10.1002/num.22734 Ziting Liu 1 , Qi Wang 1
Affiliation
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.
中文翻译:
空间分数阶对流扩散方程的非标准有限差分方法
本文提出了一种非标准的有限差分方案来求解空间分数阶对流扩散方程。通过使用傅里叶-冯·诺依曼方法,我们证明了非标准有限差分方案是无条件稳定的。我们进一步讨论数值方法的收敛性并给出收敛的顺序。数值算例表明,与经典数值方法相比,非标准有限差分法可以有效地减小最大误差并提高数值求解的精度。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,非标准有限差分方案对于求解分数阶偏微分方程是可行且有效的。
更新日期:2020-12-23
中文翻译:
空间分数阶对流扩散方程的非标准有限差分方法
本文提出了一种非标准的有限差分方案来求解空间分数阶对流扩散方程。通过使用傅里叶-冯·诺依曼方法,我们证明了非标准有限差分方案是无条件稳定的。我们进一步讨论数值方法的收敛性并给出收敛的顺序。数值算例表明,与经典数值方法相比,非标准有限差分法可以有效地减小最大误差并提高数值求解的精度。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,非标准有限差分方案对于求解分数阶偏微分方程是可行且有效的。