Algorithm Development, Analysis, and Application of High Order Schemes

在这个项目中，PI将研究求解偏微分方程的高阶精确和高效数值方法的算法设计和分析。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。PI计划研究的算法包括有限差分和有限体积加权基本非振荡（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。提出的研究主题将包括研究笛卡尔网格有限差分格式的高阶数值边界条件的反Lax-Wendroff过程，解决一般几何问题，多物质流动的拉格朗日型有限体积格式，具有强激波的不连续Galerkin方法的简单加权本质非振荡限制器，任意点云的高阶稳定保守方法，弱耦合双曲型多域和网络问题的间断Galerkin方法，间断Galerkin格式的高效时间步进技术，高阶精确保界格式及其应用，辐射传递方程的保界高阶间断Galerkin格式，Drude超材料中Maxwell方程的节能DG方法，带障碍物前传播问题的高效间断Galerkin方法，非连续Galerkin方法的超收敛分析及其应用，基于非连续Galerkin框架的多尺度方法，以及计算生物学中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 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.