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PDE开发准确，高效且鲁棒的数值方法，特别是提及涉及非平滑（不连续的）解决方案和问题的问题，涉及高度分散的尺度，因此很难在数值上求解。拟议研究的主要部分将集中于解决涉及复杂非线性波现象的新技术的发展，复杂的计算领域的问题，移动边界和材料/层界面以及包括不确定现象的问题。新技术将基于粒子方法和有限体积方法及其杂交。后一种方法将利用粒子方法的主要优势作为无网状方法和捕捉有限体积的方法，尤其是在复杂几何，自由边界和结构相互作用的流动问题中。随机工具和数值工具的组合也将用于解决不确定性和问题的问题，其中应考虑多个量表。该设计的方法将应用于各种非线性问题，其中包括Euler-Poincare方程，多相和多流体流量模型，在湍流不可压缩流中污染物的传输模型，趋化性模型，反应性Euler方程描述了僵硬的衰减波，零散射范围，零扩散 - 散布局部限制范围的保护范围，以及其他。随机的初始值问题，例如随机扰动的KDV方程和带有随机力的汉堡方程，也将通过提出的方法解决。重要的是，除了提供证实分析方法的示例外，上述应用对于在现代科学中引起的广泛问题具有实质性的独立价值。近年来，解决部分微分方程的数值方法已演变为一种重要且极其有效的工具，用于在不同应用的ARE中对许多现象进行定量和定性研究，否则根本无法研究这些方法。拟议的项目将有助于开发计算方法，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，在设计数字方法上的强烈重音是为复杂问题（例如多相和多流体模型，污染传播的模型，聚合物系统，聚合物系统，趋化模型，主动流体传输模型，多规模和随机的初始价值问题等）的设计。多组分流，颗粒流，反应性流动，聚合物流和其他领域。研究方法的广泛应用也反映了拟议项目的跨学科特征。