Abstract This paper introduces a high-order numerical procedure to solve the two-dimensional distributed-order Riesz space-fractional diffusion equation. In the proposed technique, first, a second-order numerical integration rule is employed to estimate the integral of the distributed-order Riesz space-fractional derivative. Then, the time derivative is discretized by a second-order difference scheme. Finally, the spatial direction is approximated by a difference formulation with fourth-order accuracy. The stability of the semi-discrete scheme is analyzed. We conclude that the difference between two consecutive time steps i.e. U i , j n − U i , j n − 1 is nearly zero when n → ∞ . So, a suitable term is added to the main difference scheme as by adding this term we could derive the main ADI scheme. Furthermore, to reduce the used CPU time, we combine the fourth-order ADI formulation with the proper orthogonal decomposition method and then we gain a POD based reduced-order compact ADI finite difference plane. In the next, the convergence order of the fully discrete formulation has been investigated. The numerical results show the efficiency of new technique. It must be noted that the finite difference method is an effective and robust numerical technique for solving nonlinear equations that the ADI approach can be combined with it to improve the numerical simulations.

中文翻译：

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求解二维分布阶Riesz空间分数扩散方程的基于POD的降阶Crank-Nicolson/四阶交替方向隐式(ADI)有限差分格式

摘要 本文介绍了求解二维分布阶Riesz空间分数扩散方程的高阶数值方法。在所提出的技术中，首先，采用二阶数值积分规则来估计分布式 Riesz 空间分数阶导数的积分。然后，时间导数通过二阶差分格式离散化。最后，空间方向由具有四阶精度的差分公式逼近。分析了半离散方案的稳定性。我们得出结论，当 n → ∞ 时，两个连续时间步长之间的差异，即 U i , jn − U i , jn − 1 几乎为零。因此，一个合适的项被添加到主要差异方案中，因为通过添加这个项，我们可以推导出主要的 ADI 方案。此外，为了减少使用的 CPU 时间，我们将四阶 ADI 公式与适当的正交分解方法相结合，然后我们获得了基于 POD 的降阶紧凑 ADI 有限差分平面。接下来，研究了完全离散公式的收敛阶次。数值结果表明了新技术的有效性。必须指出，有限差分法是求解非线性方程的有效且稳健的数值技术，可以将 ADI 方法与其结合以改进数值模拟。