High Order Schemes for Hyperbolic and Convection-dominated Problems

在本项目中，PI将在高阶数值方法的算法设计和分析方面进行研究。这些算法用于解决航空航天工程、半导体器件设计、天体物理和生物问题等不同应用领域中出现的科学和工程问题。即使在当今速度很快的计算机上，设计高效可靠的算法以获得这些应用问题的准确解决方案仍然是必不可少的。拟议活动产生的更广泛影响将是一套强大的计算工具，适用于上述各种应用程序。这些工具有望为这些应用中复杂解结构的计算机模拟做出积极贡献。所研究的算法包括求解双曲型和其他对流控制偏微分方程组的有限差分和有限体积加权本质无振荡格式(WENO)和间断Galerkin有限元方法。虽然本项目的重点是算法设计和分析，但将密切关注应用。研究内容包括：高阶精度保界算法及其应用的研究，物理边界与网格不对齐的矩形网格有限差分格式高阶数值边界条件的逆Lax-Wendroff方法，非保守问题的亚格子WENO格式，多物质流动的拉格朗日型有限体积格式，波动问题长期模拟的能量守恒间断Galerkin方法，求解有障碍物的前沿传播问题的高效间断Galerkin方法，间断Galerkin方法的超收敛分析及其在自适应计算中的应用，非结构网格中强激波问题间断Galerkin方法的简单WENO限制器，基于不连续Galerkin框架的多尺度方法，交通和行人流动模型的分析和数值解，宇宙学中的湍流模拟，计算生物学中的聚集和协调运动研究。应用程序中的问题将激励设计新算法或现有算法中的新功能；使用数学工具分析这些算法，以指导其适用性和局限性；解决包括并行实现问题在内的实际考虑，以使算法在大规模计算中具有竞争力；与工程师和其他应用科学家的合作使这些新算法或现有算法中的新功能能够有效应用。

In this project, the PI will perform research in the algorithm design and analysis of high order numerical methods. These algorithms are used to solve scientific and engineering problems arising from diverse application fields such as aerospace engineering, semi-conductor device design, astrophysics, and biological problems. Even with today's fast computers, it is still essential to design efficient and reliable algorithms which can be used to obtain accurate solutions to these application problems. The broader impacts resulting from the proposed activity will be a suite of powerful computational tools, suitable for various applications mentioned above. These tools are expected to make positive contributions to computer simulations of the complicated solution structure in these applications. The algorithms to be investigated include the finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes and discontinuous Galerkin finite element methods, for solving hyperbolic and other convection dominated partial differential equations (PDEs). While the emphasis of this project is on algorithm design and analysis, close attention will be paid to applications. Topics of proposed investigations will include the study on high order accurate bound-preserving algorithms and applications, an inverse Lax-Wendroff procedure for high order numerical boundary conditions for finite difference schemes on rectangular meshes when the physical boundary is not aligned with the meshes, WENO schemes with subcell resolution for nonconservative problems, Lagrangian type finite volume schemes for multi-material flows, energy-conserving discontinuous Galerkin methods for long time simulation of wave problems, efficient discontinuous Galerkin methods for front propagation problems with obstacles, superconvergence analysis of discontinuous Galerkin methods and its applications in adaptive computation, simple WENO limiters for discontinuous Galerkin methods in unstructured meshes for problems with strong shocks, multi-scale methods based on the discontinuous Galerkin framework, analysis and numerical solutions for traffic and pedestrian flow models, turbulence simulation in cosmology, and study on aggregation and coordinated movement in computational biology. Problems in applications will motivate the design of new algorithms or new features in existing algorithms; mathematics tools are used to analyze these algorithms to give guidelines for their applicability and limitations; practical considerations including parallel implementation issues are addressed to make the algorithms competitive in large scale calculations; and collaborations with engineers and other applied scientists enable the efficient application of these new algorithms or new features in existing algorithms.