A simple and fast finite difference method for the variable-order nonlocal Laplacian

主讲人：郝朝鹏 东南大学研究员

时间：2025年4月23日16：00

地点：徐汇校区三号楼332报告厅

举办单位：数理学院

主讲人介绍：郝朝鹏于2017年东南大学计算数学博士毕业，后于2020年获得伍斯特理工学院数学科学博士学位。2020年8月-2023年5月曾任普渡大学Golumb访问助理教授，2023年7月至今担任东南大学研究员，研究领域为科学计算和数值分析。现研究兴趣包括非局部偏微分方程和随机微分方程数值解。目前共发表论文三十篇左右。其中多篇文章发表在计算数学国际知名期刊MCOM，SINUM， SISC，SIAM UQ，JCP等。

内容介绍：The constant order fractional Laplacian has been extensively studied in the literature in the past decades. However, it may be insufficient for the heterogeneous effect due to the spatial variability of a complex medium. To account for heterogeneity, the variable-order operators depending on the spatial location variable have been alternatively proposed. Changing directly the constant-order into the variable-order may increase not only the model’s capability but also the complexity of computation. For the fractional Laplacian, an efficient and accurate numerical evaluation in multi-dimension is challenging due to the nature of a singular integral. To overcome this challenge, in our previous work (Hao et al. JCP 2021), we propose a simple and easy-to-implement finite difference scheme for the multi-dimensional fractional Laplacian defined by a hypersingular integral. In this talk, we extend this method to the variable-order case and propose an efficient method for the variable-order fractional Laplacian. We prove that the scheme is of second-order convergence and apply the developed finite difference scheme to solve various equations with the variable-order fractional Laplacian. We present a fast algorithm for computing the variable-order fractional Laplacian. Several numerical examples demonstrate the accuracy and efficiency of our algorithm and verify our theory.