二维空间分数阶扩散方程的高阶快速算法

High Order Fast Algorithm for Fractional Diffusion Equations in Two-dimensional Space

分数阶扩散方程广泛的应用于力学、生物学、金融学、图像处理等领域。由于分数阶微分算子具有很大的稠密性,针对二维的空间分数阶扩散方程,采用了高阶数值方法进行离散,并且使用2个参数的Kronecker积分裂迭代预处理使其更快收敛。通过数值实验表明,边值方法计算的收敛阶达到四阶精度,并且双参数的Kronecker积分裂迭代预处理效果比直接计算大大减少了计算成本。说明该算法具有高精度和计算的优势性。

Fractional diffusion equations are widely used in the fields of mechanics,biology,finance,image processing and so on.Due to the high density of fractional differential operators,a high order numerical method is used to discretethe two-dimensional spatial FDEs,and a two-parameter Kronecker product splitting iterative preprocessing is used to make it converge faster.The numerical experiments show that the convergence order of the boundary value method reaches the fourth order precision,and the two-parameter Kronecker product splitting iterative preprocessing effect greatly reduces the calculation cost than the direct calculation.It shows that the algorithm has the advantages of high precision and calculation.