High Order Schemes for Hyperbolic and Convection-dominated Problems

在这个项目中，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。 即使有今天的快速计算机，它仍然是必不可少的设计有效和可靠的算法，可用于获得这些应用问题的准确解决方案。 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。 这些工具预计将作出积极的贡献，在这些应用程序中的复杂的解决方案结构的计算机模拟。研究的算法包括有限差分和有限体积加权基本无振荡（韦诺）格式和间断Galerkin有限元方法，用于求解双曲型和其他对流占优偏微分方程（PDE）。虽然这个项目的重点是算法的设计和分析，密切关注的应用程序。研究内容包括高精度边界保持算法及其应用、物理边界与网格不一致时矩形网格上有限差分格式的高精度数值边界条件的Lax-Wendroff逆算法、非保守问题的子单元韦诺格式、多物质流的拉格朗日型有限体积格式、能量守恒间断Galerkin方法在波动问题长时间模拟中的应用，间断Galerkin方法在有障碍物的波前传播问题中的应用，间断Galerkin方法的超收敛性分析及其在自适应计算中的应用，间断Galerkin方法在强激波问题非结构网格中的简单韦诺限制器，基于不连续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，以及计算生物学中的聚集和协调运动研究。 应用中的问题将激发新算法的设计或现有算法的新功能;数学工具被用来分析这些算法，以给出它们的适用性和局限性的指导方针;包括并行实现问题在内的实际考虑被解决，以使算法在大规模计算中具有竞争力;并且与工程师和其他应用科学家的合作使得能够有效地应用这些新算法或现有算法中的新特征。

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.