讲座题目：A meshfree collocation method based on moving Taylor polynomial approximation

主办单位：三峡数学研究中心/理学院

报告专家：王晓东 教授（西北工业大学）

报告时间：2021年12月4日（周六） 上午（9:00—12: 00）

报告地点：腾讯会议（ID： 730 238 7023；密码：123321）

专家简介：王晓东，西北工业大学数学与统计学院教授，从事计算数学和应用数学的教学和科研工作。近年来，主要围绕高性能聚合物制品精密制造中的宏观成形和介观成性两大问题开展数学建模与高性能计算研究。研究方向涉及偏微分方程数值求解的高性能算法、粘弹流动模拟的稳定数值格式、聚合物结晶问题的数学建模、气固两相系统的可计算建模等。先后主持国家自然科学基金面上项目2项、青年项目1项、省部级课题4项，并参与国家重大科研项目3项。注重与企业和科研院所之间的合作，主持完成横向课题7项，促进了数学研究成果走向落地。在Int J Heat Mass Tran, J Chem Phys, Soft Matter, J Sci Comput等期刊上发表论文40余篇。获陕西省优秀博士论文奖、陕西省高等学校科学技术一等奖(第1完成人)、军队科技进步三等奖等。入选陕西省高校科协青年人才托举计划。

报告摘要： This talk presents a meshfree collocation method for solving high order partial differential equations (PDEs). The leading numerical difficulty is the approximation of high order derivatives. To make the approximation simple and efficient, a moving Taylor polynomial (MTP) approximation is presented by using movable expansion point for each sub-domain. Derivatives can be derived straightforward from the corresponding Taylor coefficients, which are determined by solving a weighted least squares problem. A distinct feature of the method is its ability to give the derivatives along with the shape function itself without further cost. To ensure the accuracy of high order approximation, stability of the weighted least squares problems for determining the Taylor coefficients is another issue should be addressed. For this purpose, the basis functions are rescaled by the size of window functions, and QR decomposition is adopted to solve the weighted least squares problems. The collocation method based on this MTP approximation does not require any grid or background cell, so it is a truly meshfree method. When solving the linear algebraic system generated by the MTP collocation method, a preconditioned sparse biconjugate gradients stabilized (BICGSTAB) solver is used to accelerate the computation speed. Numerical tests show that the proposed method is much accurate and efficient for high order PDEs.