Particle Methods for Nonlinear Time-Dependent PDEs

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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.