Abstract:

When solving partial differential equations, finite difference methods have the advantage of simplicity, however they are usually only designed on Cartesian meshes.  In this talk, we will discuss a class of high order finite difference numerical boundary condition for solving hyperbolic Hamilton-Jacobi equations, hyperbolic conservation laws, and convection-diffusion equations on complex geometry using a Cartesian mesh.  The challenge results from the wide stencil of the interior high order scheme and the fact that the boundary may not be aligned with the mesh. Our method is based on an inverse Lax-Wendroff procedure for the inflow boundary conditions coupled with traditional extrapolation or weighted essentially non-oscillatory (WENO) extrapolation for outflow boundary conditions.  The schemes are shown to be high order and stable, under the standard CFL condition for the inner schemes, regardless of the distance of the first grid point to the physical boundary, that is, the ``cut-cell'' difficulty is overcome by this procedure.  Recent progress in nonlinear conservation laws with sonic points, and a conservative version of the method, will be discussed. Numerical examples are provided to illustrate the good performance of our method.

Bio:

Chi-Wang Shu obtained his BS degree from the University of Science and Technology of China in 1982 and his PhD degree from UCLA in 1986.  He has been at Brown University since 1987, where he is the Chair of the Division of Applied Mathematics since 2023 and between 1999 and 2005, and is the Theodore B. Stowell University Professor of Applied Mathematics. His research interest includes high order numerical methods for solving hyperbolic and other convection dominated PDEs, with applications in CFD and other areas. He is the Chief Editor of Journal of Scientific Computing and of Communications on Applied Mathematics and Computation, and serves in the editorial boards of several other journals including Journal of Computational Physics and Mathematics of Computation.  He is a SIAM Fellow, an AMS Fellow and an AWM Fellow, and an invited 45-minute speaker in the International Congress of Mathematicians (ICM) in 2014. He received the First Feng Kang Prize of Scientific Computing in 1995, the SIAM/ACM Prize in Computational Science and Engineering in 2007, and the SIAM John von Neumann Prize in 2021.  

Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Inverse_Lax_Wendroff_Procedure_for_Numerical_Boundary_Conditions.html