计算高维带弱奇异核发展型方程的交替方向隐式欧拉方法

本文主要研究高维带弱奇异核的发展型方程的交替方向隐式(ADI)差分方法.向后欧拉(Euler)方法联立一阶卷积求积公式处理时间方向的离散,有限差分方法处理空间方向的离散,并进一步构造了ADI全离散差分格式.然后将二维问题延伸到三维问题,构造三维空间问题的ADI差分格式.基于离散能量法,详细证明了全离散格式的稳定性和误差分析.随后给出了2个数值算例,数值结果进一步验证了时间方向的收敛阶为一阶,空间方向的收敛阶为二阶,和理论分析结果一致.

ALTERNATING DIRECTION IMPLICIT EULER METHOD FOR THE HIGH-DIMENSIONAL EVOLUTION EQUATIONS WITH WEAKLY SINGULAR KERNEL

In this paper, an alternating direction implicit (ADI) difference method is considered for the high dimensional evolution equations with the weak singular kernel. The backward Euler method and the first-order convolution quadrature formula are used in time direction, finite difference method is used in space direction, and then the ADI fully discrete scheme is obtained. The stability and error analysis are proved by the discrete energy method. We extend the two-dimensional problem to the three-dimensional problem and construct the ADI difference scheme for the three-dimensional space problem. Finally, two numerical examples are given. The numerical results show that the convergence order of our scheme is order one in time direction and order two in space direction, which is consistent with the theoretical analysis results.