高维时间分数阶扩散方程的快速高效数值格式

A Fast and Efficient Numerical Scheme for High-Dimensional Time-Fractional Order Diffusion Equations

摘要: 本文研究了关于高维时间分数阶扩散方程的高效数值算法问题。提出了两种数值格式，一种为改进的L2数值格式，该格式有效地解决了普通L2格式在初始时刻会出现收敛精度下降的问题，并在空间方向上运用紧致差分算子进行离散，使得该格式的收敛精度达到时间上$3-\alpha$ 阶，空间上4阶。另一种在改进L2格式的基础上，运用指数和方法去逼近时间分数阶导数中的幂函数，从而极大地减少了计算成本，并且保持同样的高收敛精度。针对具有初值奇异性的问题，我们采用梯度网格方法去解决。最后通过多个数值结果证明了这两种算法的有效性和准确性，并展示所提出快速化算法的计算效率。

Abstract: In this paper, the problem of efficient numerical algorithms on high-dimensional time fractional order diffusion equations is investigated. Two numerical schemes are proposed, one is the improved L2 numerical scheme, which efficiently solves the problem of decreasing convergence accuracy at the initial moment of the ordinary L2 scheme, and discretizes it in the spatial direction by applying the compact difference operator, so that the convergence accuracy of this scheme reaches the order in time and the order in space. The other is to improve the L2 scheme by applying the sum of exponentials method to approximate the power function in the fractional order derivatives in time, which greatly reduces the computational cost and maintains the same high convergence accuracy. For problems with initial value singularities, we use the graded meshes method to solve them. Finally, the effectiveness and accuracy of both algorithms are demonstrated through several numerical results and the computational efficiency of the proposed fastening algorithm is shown.