报告题目：Positivity Preserving Limiters for Time-Implicit Discontinuous Galerkin discretizations

报 告 人：J.J.W. van der Vegt

Department of Applied Mathematics, University of Twente, Enschede, The Netherlands

报告时间： 2024年10月8日（星期二）下午16：00

报告地点： 文理楼290

报告人简介：

J.J.W. van der Vegt，荷兰University of Twente大学教授，中国科学技术大学客座教授. 1988年博士毕业于荷兰Delft大学，于Stanford大学做博士后，1991年开始先后工作于NASA 研究中心，荷兰国家宇宙空间实验室，荷兰University of Twente大学。获安徽省黄山友谊奖。担任Journal of Scientific Computing，Communications on Applied Mathematics and Computation等期刊主编。主要研究方向为间断有限元方法的理论及其应用，高精度数值方法，保界算法等数值方法。在业内顶尖期刊SIAM Journal on Scientific Computing, Journal of Computational Physics 等发表60余篇科研论文。

报告摘要： 

In this presentation an overview will be given of a novel approach to ensure positivity and bounds preservation of Local Discontinuous Galerkin (LDG) discretizations coupled with implicit time discretization methods. Most currently existing positivity preserving numerical discretizations can only be combined with explicit time integration methods. Both the chemically reactive Euler equations and a class of incompletely parabolic partial differential equations will be considered.

To ensure positivity and bounds preservation of the numerical solution, we use the Karush-Kuhn-Tucker (KKT) limiter, which imposes these bounds explicitly by coupling these conditions with time-implicit LDG discretizations using Lagrange multipliers. This results in the well-known Karush-Kuhn-Tucker equations, which are solved using a semi-smooth Newton method. First, the basic algorithm will be explained for incompletely parabolic partial differential equations and demonstrated on some models problems. Next, we will consider the chemically reactive Euler equations. To account for the large disparity between the convective and chemical time scales in the chemically reactive Euler equations, a second order Strang operator splitting approach is used to split these equations into the homogeneous Euler equations and a reaction equation. The KKT limiter to ensure positivity and bounds preservation is then applied to both equations. Special attention will be given to the proper treatment of the chemical reactions in shock and detonation regions since an inaccurate position of discontinuities can strongly influence the chemical reactions and result in spurious numerical solutions.

Numerical results will be presented to demonstrate the accuracy and positivity preservation for several model problems, including chemical reactions, shocks and detonations, that require a positivity and bounds preserving limiter.