一、题目

A new Lagrange multiplier approach for constructing positivity/bound preserving schemes

二、主讲人

Jie Shen

三、摘要

Solutions for a large class of partial differential equations (PDEs) arising from sciences and engineering applications are required to be positive to be positive or within a specified bound. It is of critical importance that their numerical approximations preserve the positivity/bound at the discrete level, as violation of the positivity/bound preserving may render the discrete problems ill posed. I will review the existing approaches for constructing positivity/bound preserving schemes, and then present a new Lagrange multiplier approach for constructing a class of positivity/bound preserving schemes for parabolic type equations. The new approach introduces a space-time Lagrange multiplier to enforce the positivity/bound using the Karush-Kuhn-Tucker (KKT) conditions. We then use a predictor-corrector approach to construct a class of positivity/bound preserving schemes: with a generic semi-implicit or implicit scheme as the prediction step, and the correction step, which enforces the positivity/bound preserving, can be implemented with negligible cost. We shall present some stability/error analysis for our schemes under a general setting, and present ample numerical results to validate the new approach.

四、主讲人简介

沈捷，美国普渡大学数学系教授、国际著名数值计算和分析专家。1982年毕业于北京大学计算数学专业，随后赴法国巴黎十一大学研究数值分析，师从国际著名数学大师ROGER TEMAN。沈捷教授长期从事偏微分方程数值解研究，尤其在谱方法和投影法上有很多杰出的工作，目前已在SIAM Review, SIAM J. Numer. Anal.,SIAM J. Sci. Comput.,Numer. Math., Math. Comp.等国际著名期刊上发表学术论文200余篇，其研究结果被国际同行广泛引用。2008年沈捷教授因在微分方程研究中的卓越贡献获得富布赖特奖，国家级高层次特聘专家。2017年当选美国数学会Fellow, 2020年当选国际工业与应用数学协会(SIAM)Fellow。

五、邀请人

芮洪兴 数学学院教授

六、时间

10月23日（周六）9:00

七、地点

中心校区知新楼B座924报告厅

八、主办方

山东大学数学学院