Efficient High Order Numerical Methods for Convection Dominated Partial Differential

在本项目中，将研究双曲型和其他对流主导的偏微分方程，特别是在自适应、多尺度和不确定环境下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非振荡（WENO）格式、不连续Galerkin有限元方法和粒子方法。还将讨论这些方法的并行实现和应用。所提出的活动的智力价值在于它对算法开发、分析、实现和应用的全面覆盖。应用中的问题激发了新算法的设计或现有算法的新特性；数学工具被用来分析这些算法，为它们的适用性和局限性提供指导；包括并行实现问题在内的实际考虑被解决，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可以有效地应用这些新算法或现有算法的新功能。本研究旨在设计有效的演算法，当这些演算法应用于当今强大的计算机时，将有助于解决飞机设计的空气动力学和空气声学，通信的电磁波模拟以及计算机工业的半导体器件模拟等各种应用中的许多问题。这个建议的主旨是使用强大的数学工具来指导算法的设计，使它们在应用中更有效，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.