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Positivity Preserving Limiters for Time-Implicit Higher Order Discontinuous Galerkin Discretizations

  • 演讲者:徐岩(中国科学技术大学)

  • 时间:2022-11-24 16:00-17:00

  • 地点:Tencent Meeting ID 227 933 421, Passcode 666666

Abstract 

In this talk, we present higher-order bounds preserving time-implicit discontinuous Galerkin (DG) discretizations for the nonlinear degenerate parabolic equations and the reactive Euler equations. Using Lagrange multipliers the conditions imposed by the positivity preserving limiters are directly coupled to a DG discretization combined with a Diagonally Implicit Runge-Kutta time integration method. The positivity preserving DG discretization is then reformulated as a Karush-Kuhn-Tucker (KKT) problem, which is frequently encountered in constrained optimization. Since the limiter is only active in areas where positivity must be enforced it does not affect the higher-order DG discretization elsewhere. The resulting non-smooth nonlinear algebraic equations have, however, a different structure compared to most constrained optimization problems. We develop an efficient active set semi-smooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations. Convergence of this semi-smooth Newton method is proven using a specially designed quasi-directional derivative of the time-implicit positivity preserving DG discretization. Numerical results are shown to demonstrate that the bounds preserving DIRK-DG discretizations are higher order accurate for smooth solutions and also efficient for stiff problems with discontinuities. 


报告人简介: 
徐岩,中国科学技术大学数学科学学院教授。2005 年于中国科学技术大学数学系获计算数学博士学位。2005-2007 年在荷兰 Twente 大学从事博士后研究工作。2009 年获得德国洪堡基金会的支持在德国 Freiburg 大学访问工作一年。主要研究领域为高精度数值计算方法。2008 年度获全国优秀博士学位论文奖,2017 年获国家自然科学基金委“优秀青年基金”。徐岩教授入选了教育部新世纪优秀人才计划,主持国家自然科学基金面上项目、德国洪堡基金会研究组合作计划(Research Group Linkage Programme)、霍英东青年教师基础研究课题等科研项目。担任Journal of Scientific Computing, Advances in Applied Mathematics and Mechanics, Communication on Applied Mathematics and Computation、计算物理等杂志的编委。