汇报标题 (Title)：High-Order Implicit Shock Tracking(高阶隐式冲击跟踪)
汇报人 (Speaker)：Andrew Shi博士后（上海纽约大学）
汇报功夫 (Time)：2025年3月20日(周四) 15:30-16:30
汇报地址 (Place)：校本部GJ303
约请人(Inviter)：李常品、蔡敏
主办部门：理学院数学系
汇报提要：We present a framework for resolving discontinuous solutions of conservation laws using implicit tracking and a high-order discontinuous Galerkin (DG) discretization. Central to the framework is an optimization problem and associated sequential quadratic programming solver which simultaneously solves for a discontinuity-aligned mesh and the corresponding high-order approximation to the flow that does not require explicit meshing of the a priori unknown discontinuity surface. We utilize an error-based objective function that penalizes violation of the DG residual in an enriched test space, which endows the method with r-adaptive behavior: mesh nodes move to track discontinuities with element faces and improve the conservation law approximation in smooth regions of the flow. This method is shown to deliver highly accurate solutions on coarse, high-order discretizations without nonlinear stabilization and recover optimal convergence rates O(h^{p+1}) for problems with discontinuous solutions. We demonstrate this framework on a series of inviscid steady and unsteady conservation laws, the latter of which using both a space-time and method of lines discretization.
