Abstract: Traditional time discretization schemes  are usually based on Taylor expansions at $t_{n+\beta}$ with $\beta\in [0,1]$. However, their ability to deal with very stiff problems are limited by their stability regions. Furthermore, their stability regions decrease as their order of accuracy increase. We show that by using Taylor expansion at $t_{n+\beta}$ with $\beta>1$ as a parameter, we can construct a new class of schemes whose stability region increases with $\beta$, thus allowing us to choose $\beta$ according to the stability and accuracy requirement. 

In addition, this approach enabled us to solve a long standing problem in the numerical approximation of Navier-Stokes equations. More precisely, no decoupled scheme for the time-dependent Stokes problem with second- or higher-order pressure extrapolation were proven to be unconditionally stable. By choosing suitable $\beta$, we were able to construct unconditionally stable (in $H^1$ norm), decoupled consistent splitting schemes up to fifth-order for the time-dependent Stokes problem. 

Then, by combining the generalized SAV approach with the new consistent splitting schemes, we were able to construct  unconditionally stable  and totally decoupled schemes of second- to fourth order with uniform  optimal error estimates. We shall also  present ample numerical results to show the computational advantages of these schemes.

报告人简介：沈捷教授于1982年毕业于北京大学计算数学专业, 1983年公派赴法国巴黎十一大学留学, 于1987年获得博士学位后赴美国Indiana University从事博士后研究。 1991年至2001年先后任美国 Pennsylvania State University 数学系助理教授，副教授，教授。 2002年起任美国普度（Purdue) 大学数学系教授，2012年至2022年任普度大学计算与应用数学中心主任, 2023年被授予普度大学杰出教授。

沈捷教授在国际杂志上发表论文250多篇，并有两本专著, 其研究结果被国际同行广泛引用，在Google Scholar上被引用逾两万五千次。 他于2009年被聘国家级高层次人才，2017年当选美国数学会 （AMS） Fellow， 2020年当选国际工业与应用数学协会（SIAM）Fellow。