Efficient High Order Numerical Methods for Convection Dominated Partial Differential

本项目将研究双曲型和其他对流主导偏微分方程的高阶数值方法的算法设计和分析，包括有限差分和有限体积加权基本无振荡（韦诺）格式、间断Galerkin有限元方法和粒子方法，特别是在自适应、多尺度和不确定环境中。 这些方法的并行实现和应用也将得到解决。 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。 应用中存在的问题激发了新算法或现有算法中新特性的设计;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提出，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。本文提出的研究目标是设计高效的算法，当这些算法在当今功能强大的计算机上使用时，将有助于解决来自各种应用的许多问题，例如用于飞行器设计的空气动力学和航空声学问题，用于通信的电磁波模拟，以及用于计算机工业的半导体器件模拟。本文的主旨是使用强大的算法工具来指导算法的设计，使它们在应用中更高效、更可靠、更健壮。

In this project, research in the algorithm design and analysisof high order numerical methods, including the finite differenceand finite volume weighted essentially non-oscillatory (WENO) schemes, discontinuous Galerkin finite element methods, and particle methods, for hyperbolic and other convection dominated partial differential equations, especially in adaptive, multiscale and uncertain environments, will be carried out. Parallel implementation and applications of these methods will also be addressed. The intellectual merit of the proposed activity lies in its comprehensive coverage of algorithm development, analysis,implementation and applications. Problems in applicationsmotivate the design of new algorithms or new features inexisting algorithms; mathematics tools are used to analyzethese algorithms to give guidelines for their applicabilityand limitations; practical considerations including parallelimplementation issues are addressed to make the algorithmscompetitive in large scale calculations; and collaborationswith engineers and other applied scientists enable theefficient application of these new algorithms or new featuresin existing algorithms.The proposed research aims at the design of efficient algorithms,which, when used on today's powerful computers, will help to solve many problems from diversified applications such asaerodynamics and aeroacoustics for aircraft design, electromagnetism wave simulation for communications, and semiconductor device simulation for the computer industry.The thrust of this proposal is to use powerful mathematicaltools to guide the design of algorithms, so that they are moreefficient, more reliable, and more robust in applications.