講座主題：A new Lagrange multiplier approach for constructing positivity/bound preserving schemes
專(zhuān)家姓名：沈捷
工作單位：美國(guó)普渡大學(xué)
講座時(shí)間：2021年10月26日16:00-17:00
講座地點(diǎn)：數(shù)學(xué)院會(huì)議室341
主辦單位：煙臺(tái)大學(xué)數(shù)學(xué)院
內(nèi)容摘要：
Solutions for a large class of partial differential equations (PDEs) arising from sciences and engineering applications are required 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.
主講人介紹：
沈捷教授于1982年畢業(yè)于北京大學(xué)計(jì)算數(shù)學(xué)專(zhuān)業(yè), 1983年公派赴法國(guó)巴黎十一大學(xué)留學(xué),于1987年獲得博士學(xué)位后赴美國(guó)Indiana University從事博士后研究。1991年至2001年先后任美國(guó)Pennsylvania State University數(shù)學(xué)系助理教授，副教授，教授。2002年起任美國(guó)普度大學(xué)數(shù)學(xué)系教授，2012年起任普度大學(xué)計(jì)算與應(yīng)用數(shù)學(xué)中心主任。目前擔(dān)任8個(gè)國(guó)際雜志的編委。沈捷教授主要從事偏微分方程數(shù)值解研究工作，具體研究方向包括譜方法數(shù)值分析理論,計(jì)算流體,以及計(jì)算材料科學(xué)。在國(guó)際雜志上發(fā)表論文200多篇，并有兩本專(zhuān)著,其研究結(jié)果被國(guó)際同行廣泛引用，在Google Scholar上被引用逾一萬(wàn)七千次。 他于2009年被聘為教育部長(zhǎng)江講座教授，2017年當(dāng)選美國(guó)數(shù)學(xué)會(huì)Fellow，2020年當(dāng)選國(guó)際工業(yè)與應(yīng)用數(shù)學(xué)協(xié)會(huì)（SIAM）Fellow。
