Algorithm Development, Analysis, and Application of High Order Schemes

在这个项目中，PI将在算法设计和分析求解偏微分方程组的高精度、高效率的数值方法方面进行研究。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。PI计划研究的算法包括求解双曲和其他对流控制的偏微分方程(PDE)的有限差分和有限体积加权基本无振荡(WENO)格式和间断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。建议的研究课题包括：求解一般几何问题的笛卡尔网格有限差分格式高阶数值边界条件的反Lax-Wendroff方法，多物质流动的拉格朗日型有限体积格式，强激波间断Galerkin方法的简单加权基本无振荡限制器，任意点云的高阶稳定守恒方法，弱耦合双曲多域和网络问题的间断Galerkin方法，间断Galerkin格式的高效时间步长技术，高精度高精度保界格式及其应用，辐射传递方程的保界高阶Galerkin格式，Drude元材料中Maxwell方程的能量守恒DG方法，基于不连续Galerkin框架的多尺度方法，以及在计算生物学中的交通和行人流模型与聚集、协调运动和细胞增殖等领域的应用。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 algorithm design and analysis of high order accurate and efficient numerical methods for solving partial differential equations. 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 the PI plans to investigate 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 an inverse Lax-Wendroff procedure for high order numerical boundary conditions for finite difference schemes on Cartesian meshes solving problems in general geometry, Lagrangian type finite volume schemes for multi-material flows, a simple weighted essentially non-oscillatory limiter for discontinuous Galerkin methods with strong shocks, high order stable conservative methods on arbitrary point clouds, discontinuous Galerkin methods for weakly coupled hyperbolic multi-domain and network problems, efficient time-stepping techniques for discontinuous Galerkin schemes, high order accurate bound-preserving schemes and applications, bound-preserving high order discontinuous Galerkin schemes for radiative transfer equations, energy-conserving DG methods for Maxwell's equations in Drude metamaterials, efficient discontinuous Galerkin method for front propagation problems with obstacles, superconvergence analysis of discontinuous Galerkin methods and its applications, multi-scale methods based on the discontinuous Galerkin framework, and applications in areas including traffic and pedestrian flow models and aggregation, coordinated movement and cell proliferation 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.