Particle Methods for Nonlinear Time-Dependent PDEs

研究者开发了非线性偏微分方程的无网格粒子方法，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，以适应不同的平滑的解决方案。其他目标是发展混合有限体积-粒子方法，提高粒子方法的准确性和效率。这些方法应用于几个说明性问题：守恒定律的零扩散-扩散极限、欧拉-庞加莱方程、多相和多流体流动模型、不连续系数的污染物输运问题、趋化细菌聚集模拟和随机初值问题。用数值方法求解偏微分方程是对不同应用领域的许多现象进行定量和定性研究的一个重要而有效的工具，否则这些现象根本不可能被研究。研究者开发了一些数值方法，这些方法在计算偏微分方程的解方面比其他技术有一些优势，特别是对于复杂几何或移动边界的问题，并将这些方法应用于一些说明性的例子。

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.