Innovative Numerical Methods for Nonlinear Time-Dependent PDEs

非线性瞬态偏微分方程的创新数值方法

• 批准号: 0712898
• 负责人: Alina Chertock
• 金额: $ 27.2万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0712898&HistoricalAwards=false
• 关键词: Innovative; Numerical; Methods; Nonlinear; Time

The project is aimed at developing accurate, efficient, and robust numerical methods for nonlinear PDEs, with particular reference to problems that admit nonsmooth (discontinuous) solutions and problems that involve highly disparate scales, and therefore, are difficult to solve numerically. The principal part of the proposed research will be focused on the development of new techniques for solving problems involving complicated nonlinear wave phenomena, problems with complex computational domains, moving boundaries and material/layer interfaces, as well as problems that include uncertain phenomena. The new techniques will be based on particle methodsand finite-volume methods, as well as their hybridization. The latter approach will utilize major advantages of particle methods, as mesh-free methods, and shock-capturing finite-volume methods, especially in problems with complex geometries, free boundaries, and flows with structural interactions. A combination of stochastic and numerical tools will also be used for solving problems with uncertainties and problems, in which multiple scales should be taken into account. The designed methods will be applied to a variety of nonlinear problems, among which are the Euler-Poincare equations, multi-phase and multi-fluid flow models, models of transport of pollutant in turbulent incompressible flow, chemotaxis models, reactive Euler equations describing stiff detonation waves, zero diffusion-dispersion limits for conservation laws, and others. Stochastic initial-value problems such as the randomly perturbed KdV equation and the Burgers equation with random force will also be solved by the proposed methods. It is significant that, besides providing the examples that corroborate the analytical approach, the foregoing applications are of a substantial independent value for a broad class of problems arisingin modern science. In recent years, numerical methods for solving partial differential equations have evolved into an important and extremely efficient tool for the quantitative and qualitative study of many phenomena in different applied ares that otherwise could not have been studied at all. The proposed project will contribute significantly toward development of computational methods and will provide considerably more powerful tools for analyzing applied problems on the computer. In this proposal, a strong accent is put on designing numerical methods for complicated problems such as multi-phase and multi-fluid models, models of pollution propagation, polymer systems, chemotaxis models, active fluid transport models, multi-scale and stochastic initial-value problems, etc. These problems arise in a variety of scientific applicationsin fluid and gas dynamics, geophysics, meteorology, astrophysics, multi-component flows, granular flows, reactive flows, polymer flows, and other fields. A wide spectrum of applications of the studied methods reflects also the interdisciplinary character of the proposed project.

该项目旨在为非线性偏微分方程开发准确、高效和鲁棒的数值方法，特别是那些允许非光滑（不连续）解的问题以及涉及高度不同尺度的问题，因此难以数值求解。拟议研究的主要部分将集中于开发新技术来解决涉及复杂非线性波现象、复杂计算域、移动边界和材料/层界面的问题以及包含不确定现象的问题。新技术将基于粒子方法和有限体积方法以及它们的混合。后一种方法将利用粒子方法的主要优点，如无网格方法和冲击捕获有限体积方法，特别是在复杂几何形状、自由边界和具有结构相互作用的流动问题中。随机和数值工具的组合也将用于解决具有不确定性和需要考虑多个尺度的问题。所设计的方法将应用于各种非线性问题，其中包括Euler-Poincare方程、多相多流体流动模型、不可压缩湍流污染物传输模型、趋化性模型、描述刚性的反应性Euler方程爆炸波、守恒定律的零扩散-色散极限等。随机初值问题，例如随机扰动的 KdV 方程和具有随机力的 Burgers 方程也将通过所提出的方法来解决。值得注意的是，除了提供证实分析方法的例子之外，上述应用对于现代科学中出现的广泛问题具有重要的独立价值。近年来，求解偏微分方程的数值方法已发展成为一种重要且极其有效的工具，可以对不同应用领域中的许多现象进行定量和定性研究，而这些现象本来是无法研究的。拟议的项目将为计算方法的发展做出重大贡献，并将为分析计算机上的应用问题提供更强大的工具。在该提案中，重点关注复杂问题的数值方法设计，例如多相和多流体模型、污染传播模型、聚合物系统、趋化性模型、主动流体传输模型、多尺度和随机初始模型。这些问题出现在流体和气体动力学、地球物理学、气象学、天体物理学、多组分流、颗粒流、反应流、聚合物流等领域的各种科学应用中。研究方法的广泛应用也反映了拟议项目的跨学科特征。