讲座论坛
荔园数学讲坛: Geometric Quasilinearization for Analysis and Design of Bound-Preserving Schemes
发布时间:2022-06-30 08:09:26 1396

内容摘要: 

Solutions to many partial differential equations satisfy certain bounds or constraints. For example, the density and pressure are positive for equations of fluid dynamics, and in the relativistic case the fluid velocity is upper bounded by the speed of light, etc. As widely realized, it is crucial to develop bound-preserving numerical methods that preserve such intrinsic constraints. Exploring provably bound-preserving schemes has attracted much attention and is actively studied in recent years. This is however still a challenging task for many systems especially those involving nonlinear constraints. Based on some key insights from geometry, we systematically propose an innovative and general framework, referred to as geometric quasilinearization (GQL), which paves a new effective way for studying bound-preserving problems with nonlinear constraints. The essential idea of GQL is to equivalently transfer all nonlinear constraints into linear ones, through properly introducing some free auxiliary variables. We establish the fundamental principle and general theory of GQL via the geometric properties of convex regions, and propose three simple effective methods for constructing GQL. We apply the GQL approach to a variety of partial differential equations, and demonstrate its effectiveness and remarkable advantages for studying bound-preserving schemes, by diverse challenging examples and applications which cannot be easily handled by direct or traditional approaches.

主讲人简介: 

吴开亮,南方科技大学数学系副教授、研究员、博士生导师。2011年获华中科技大学数学学士学位;2016年获北京大学计算数学博士学位;2016-2020年先后在美国犹他大学和俄亥俄州立大学从事博士后研究;2021年1月加入南科大。研究方向包括微分方程数值解与计算流体力学、机器学习与数据驱动建模等。研究成果发表在SINUM、SISC、M3AS、Numer. Math.、J. Comput. Phys.、JSC、ApJS、Phys. Rev. D等期刊上。曾获中国数学会计算数学分会 优秀青年论文奖一等奖(2015)和中国数学会 钟家庆数学奖(2019),获得国家高层次人才计划(青年项目)和国家自然科学基金面上项目的资助。