受国际合作处资助，应数学系吴勃英教授和数学系与数学研究院孟雄副教授邀请，美国普渡大学张翔雄博士将于近日来公司进行讲学活动，欢迎感兴趣的师生参加！

报告题目1：Positivity-Preserving High Order Discontinuous Galerkin Schemes for Compressible Navier-Stokes Equations

报告时间1：2017年7月17日上午10：00—11：00

报告地点1：格物楼522

报告摘要：For gas dynamics equations such as compressible Euler and Navier-Stokes equations, preserving the positivity of density and pressure without losing conservation is crucial to stabilize the numerical computation. The L1-stability of mass and energy can be achieved by enforcing the positivity of density and pressure during the time evolution. However, high order schemes such as DG methods do not preserve the positivity. It is difficult to enforce the positivity without destroying the high order accuracy and the local conservation in an efficient manner for time-dependent gas dynamics equations. For compressible Euler equations, a weak positivity property holds for any high order finite volume type schemes including DG methods, which was used to design a simple positivity-preserving limiter for high order DG schemes in Zhang and Shu, JCP 2010. Generalizations to compressible Navier-Stokes equations are however nontrivial. We show that weak positivity property still holds for DG method solving compressible Navier-Stokes equations if a proper nonlinear penalty term is added to any finite volume or DG scheme. This allows us to obtain the first high order positivity-preserving schemes for compressible Navier-Stokes equations.

报告题目2：A Simple Curved Boundary Treatment for Explicit High Order DG Methods Solving Time-Dependent Problems

报告时间2：2017年7月17日上午11：00—12：00

报告地点2：格物楼522

报告摘要：For problems defined in a 2D domain with boundary conditions specified on a curveΓ, we consider discontinuous Galerkin (DG) schemes with high order polynomial basis on a geometry fitting triangular mesh. It is well known that directly imposing the given boundary conditions on a piece-wise segment approximation boundary ΓhΓh will render any finite element method to be at most second order accurate. Unless the boundary conditions can be accurately transferred from GΓΓ to ΓhΓh, in general curvilinear element method should be used to obtain high order accuracy. We discuss a simple boundary treatment, which can be implemented as a modified DG scheme defined on triangles adjacent to ΓhΓh. Even though integration along the curve is still necessary, integrals over any curved element are avoided. If the domain is convex, or if the true solutions can be smoothly extended to the exterior of a generic domain, the modified DG scheme is high order accurate. In these cases, numerical tests on first order and second order PDEs including hyperbolic systems and the scalar wave equation suggest that it is as accurate as the full curvilinear DG scheme if it is stable. Rigorous stability result is difficult to establish. Nonetheless, numerical tests suggest that the modified DG scheme is stable on a reasonably coarse mesh and finer ones.

报告题目3：A bound-preserving fourth order compact finite difference scheme for scalar convection diffusion equations

报告时间3：2017年7月18日上午8：00—9：00

报告地点3：格物楼503

报告摘要：We show that the classical fourth order compact finite difference approximation for convection and diffusion operators satisfies a weak monotonicity property. Based on such a property, a simple local limiter can be constructed to enforce bound-preserving without losing global conservation and fourth order accuracy. This result can be extended to nonlinear equation in multiple dimensions. Numerical results will be shown.

报告题目4：The asymptotic convergence rate of the Douglas Rachford iteration for basis pursuit

报告时间4：2017年7月19日上午9：00—10：00

报告地点4：格物楼522

报告摘要： For large scale nonsmooth convex optimization problems, first order methods involving only the subgradients are usually used thanks to their scalability to the problem size. Douglas-Rachford (DR) splitting is one of the most popular first order methods in practice. It is well-known that DR applied on dual problem is equivalent to the widely used alternating direction method of multipliers (ADMM) in nonlinear mechanics and the split Bregman method in image processing community. As motivating examples, first we will briefly review several famous convex recovery results including compressive sensing, matrix completion and Phase Lift, which represent a successful story of the convex relaxation approach attacking certain NP-hard linear inverse problems in the last decade. When DR is applied to these convex optimization problems, one interesting question of practical use is how the parameters in DR affect the performance. We will show an explicit formula of the sharp asymptotic convergence rate of DR for the simple L1 minimization. The analysis will be verified on examples of processing seismic data in Curvetlet domain. 

报告人简介：

张翔雄博士分别在2006年和2011年于中国科学技术大学数学系和美国布朗大学应用数学系获得学士和博士学位，2011年至2014年在美国麻省理工学院从事博士后课题研究，现为美国普渡大学数学系助理教授。张翔雄博士长期致力于偏微分方程高阶精度数值方法和最优化研究，在高阶加权本质无振荡方法及间断有限元方法、满足最大值原理及保正方法、凸恢复问题及最优化算法设计分析等领域取得一系列关键研究成果。张翔雄博士现主持美国国家自然科学基金1项，发表包含SIAM Journal on Numerical Analysis、SIAM Journal on Scientific Computing、Numerische Mathematik、Mathematics of Computation、Proceedings of the Royal Society A、Journal of Computational Physics在内的高水平论文20余篇。