Particle Methods for Nonlinear Time-Dependent PDEs

非线性时变偏微分方程的粒子方法

• 批准号: 0410023
• 负责人: Alina Chertock
• 金额: $ 16.09万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0410023&HistoricalAwards=false
• 关键词: Particle; Methods; Nonlinear; Time; Dependent

The investigator develops mesh-free particle methods fornonlinear partial differential equations, with particularreference to problems that admit nonsmooth (discontinuous)solutions and to problems that involve highly disparate scales. A major focus is developing procedures to recover an approximatesolution from the particle distribution that adapt to varyingsmoothness of the solution. Other aims are to develop hybridfinite-volume-particle methods and improve the accuracy andefficiency of particle methods. The methods are applied toseveral illustrative problems: zero diffusion-dispersion limitsfor conservation laws, the Euler-Poincare equations, models ofmultiphase and multifluid flows, pollutant transport problemswith discontinuous coefficients, simulation of chemotacticbacteria aggregation, snd stochastic initial value problems. Solving partial differential equations numerically is animportant and efficient tool for the quantitative and qualitativestudy of many phenomena in different applied areas that otherwisecould not have been studied at all. The investigator developsnumerical methods that offer some advantages over othertechniques for computing solutions of partial differentialequations, particularly for problems with complicated geometriesor moving boundaries, and applies the methods to a number ofillustrative examples.

研究人员开发了非线性偏微分方程的无网格粒子方法，特别是针对允许非光滑（不连续）解的问题以及涉及高度不同尺度的问题。主要焦点是开发从颗粒分布中恢复近似解决方案的程序，以适应解决方案的不同平滑度。 其他目标是开发混合有限体积粒子方法并提高粒子方法的准确性和效率。 该方法应用于几个说明性问题：守恒定律的零扩散-色散极限、欧拉-庞加莱方程、多相和多流体流动模型、具有不连续系数的污染物传输问题、趋化细菌聚集的模拟、以及随机初始值问题。 数值求解偏微分方程是定量和定性研究不同应用领域中许多现象的重要而有效的工具，否则根本无法研究这些现象。 研究人员开发了数值方法，与其他技术相比，在计算偏微分方程的解时具有一些优势，特别是对于具有复杂几何形状或移动边界的问题，并将这些方法应用于许多说明性示例。