Particle Methods for Nonlinear Time-Dependent PDEs

研究者发展了求解非线性偏微分方程组的无网格粒子方法，特别是涉及非光滑(不连续)解的问题和涉及高度不同尺度的问题。一个主要的焦点是开发从颗粒分布恢复近似解的程序，以适应解的变化和光滑性。另一个目标是发展混合有限体积质点方法，提高质点方法的精度和效率。这些方法被应用于几个例证问题：守恒律的零扩散-弥散极限，欧拉-庞加莱方程，多相多流体流动模型，具有间断系数的污染物传输问题，细菌聚集的模拟，以及随机初值问题。数值求解偏微分方程组是对不同应用领域中的许多现象进行定量和定性研究的重要而有效的工具。研究人员发展了一些数值方法来计算偏微分方程解，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一些示例性例子。

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.