High Order Schemes for Hyperbolic and Convection-dominated Problems

在这个项目中，PI将对高阶数值方法的算法设计和分析进行研究。这些算法用于解决来自不同应用领域的科学和工程问题，如航空航天工程、半导体器件设计、天体物理学和生物学问题。即使在今天的快速计算机，它仍然是必不可少的设计高效可靠的算法，可以用来获得准确的解决这些应用问题。拟议的活动所产生的更广泛的影响将是一套强大的计算工具，适用于上述各种应用。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。要研究的算法包括有限差分和有限体积加权本质非振荡（WENO）格式和不连续Galerkin有限元方法，用于求解双曲型和其他对流主导的偏微分方程（PDEs）。本项目的重点是算法设计和分析，同时也会密切关注应用。建议的研究主题将包括高阶精确保界算法和应用的研究，当物理边界与网格不对齐时，矩形网格上有限差分格式的高阶数值边界条件的反Lax-Wendroff过程，非保守问题的具有亚单元分辨率的WENO格式，多材料流动的拉格朗日型有限体积格式，长时间波问题模拟的节能型间断Galerkin方法，有障碍物前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，强冲击问题非结构网格中间断Galerkin方法的简单WENO限制，基于间断Galerkin框架的多尺度方法，交通和行人流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。应用中的问题会激发新算法的设计或现有算法的新特性；使用数学工具对这些算法进行分析，给出它们的适用性和局限性的指导；实际考虑包括并行实现问题，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能够有效地应用这些新算法或现有算法中的新功能。

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.