Algorithm Design and Analysis for High Order Numerical Methods

本项目将对求解双曲型和其他对流占优偏微分方程组的高阶数值方法的算法设计和分析进行研究，包括有限差分和有限体积加权基本无振荡格式(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，但将密切关注高效的并行实现和应用。拟议活动的智力优点在于它涵盖了算法开发、分析、实施和应用的全面内容。应用程序中的问题促使设计新算法或现有算法中的新功能；数学工具用于分析这些算法，以指导其适用性和局限性；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使这些新算法或现有算法中的新功能能够有效应用。拟议活动产生的更广泛的影响将是一套强大的计算工具，适用于在自适应、多尺度和不确定环境中涉及以对流为主的偏微分方程组的各种应用。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。应用领域包括(但不限于)计算流体动力学、交通流问题、半导体器件模拟和计算生物学。研究生将参与这个项目，并将接受关于与应用密切相关的问题的数学研究的培训。将特别注意从包括妇女在内的任职人数不足的群体招聘和培训博士生。

In this project, research in the algorithm design and analysis of high order numerical methods, including 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, will be carried out. While the emphasis of this project is on algorithm design and analysis, close attention will be paid to efficient parallel implementation and applications. The intellectual merit of the proposed activity lies in its comprehensive coverage of algorithm development, analysis, implementation and applications. Problems in applications 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 featuresin existing algorithms.The broader impacts resulting from the proposed activity will be a suite of powerful computational tools, suitable for various applications with involving convection dominated partial differential equations, in adaptive, multiscale and uncertain environments. These tools are expected to make positive contributions to computer simulations of the complicated solution structure in these applications. The application areas include (but are not limited to) computational fluid dynamics, traffic flow problems, semiconductor device simulations, and computational biology. Graduate students will be involved in this project, and will get training in performing mathematics research on problems closely related to applications. Special attention will be paid to the recruitment and training of Ph.D. students from under-represented groups including women.