Particle Methods for Nonlinear Time-Dependent PDEs

研究者开发了无网格粒子方法用于非线性偏微分方程，特别是涉及非光滑（不连续）解决方案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程序，以恢复一个近似的解决方案，从粒子分布，适应不断变化的光滑度的解决方案。 另一个目标是发展混合有限体积粒子方法，提高粒子方法的精度和效率。 这些方法被应用到几个典型的问题：零扩散-弥散守恒律极限，Euler-Poincare方程，多相和多流体流动模型，具有间断系数的污染物输运问题，化学活性细菌聚集的模拟，以及随机初值问题。 数值求解偏微分方程是定量和定性研究许多现象的一个重要而有效的工具，这些现象在不同的应用领域中是无法用其他方法研究的。 研究者开发了一些数值方法，这些方法比其他计算偏微分方程解的技术具有一些优势，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一些示例。

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.