Algorithm Design and Analysis for High Order Numerical Methods

本计划将研究高阶数值方法的演算法设计及分析，包括求解双曲型及其他对流占优偏微分方程的有限差分及有限体积加权基本无振荡（韦诺）格式及间断Galerkin有限元法。 虽然这个项目的重点是算法的设计和分析，密切关注将支付有效的并行实现和应用。拟议活动的智力价值在于其全面涵盖算法开发、分析、实施和应用。 应用中的问题激发了新算法或现有算法的新功能的设计;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;以及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的新功能得到有效应用。活动将是一套强大的计算工具，适用于各种应用，涉及对流主导偏微分方程，在自适应，多尺度和不确定的环境。 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。 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。研究生将参与该项目，并将在与应用密切相关的问题上进行数学研究方面得到培训。 将特别注意博士的招聘和培训。包括妇女在内的代表性不足群体的学生。

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.