Efficient High Order Numerical Methods for Convection Dominated Partial Differential

在本项目中，将开展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研究，特别是在自适应、多尺度和不确定环境下，包括有限差分和有限体积加权本质无振荡(WENO)格式、间断Galerkin有限元方法和质点方法。还将讨论这些方法的并行实施和应用。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。应用中的问题促使设计新的算法或现有算法中的新特性；使用数学工具对这些算法进行分析，以指导它们的适用性和局限性；解决包括并行实现问题在内的实际考虑，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的新特性得以有效应用。拟议的研究旨在设计高效的算法，当这些算法用于当今功能强大的计算机时，将有助于解决来自各种应用的许多问题，如用于飞机设计的空气动力学和空气声学，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。该建议的主旨是使用强大的数学工具来指导算法的设计，从而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.