The investigator develops mesh-free particle methods for nonlinear partial differential equations, with particular reference to problems that admit nonsmooth (discontinuous) solutions and to problems that involve highly disparate scales. A major focus is developing procedures to recover an approximate solution from the particle distribution that adapt to varying smoothness of the solution. Other aims are to develop hybrid finite-volume-particle methods and improve the accuracy and efficiency of particle methods. The methods are applied to several illustrative problems: zero diffusion-dispersion limits for conservation laws, the Euler-Poincare equations, models of multiphase and multifluid flows, pollutant transport problems with discontinuous coefficients, simulation of chemotactic bacteria aggregation, snd stochastic initial value problems.
Solving partial differential equations numerically is an important and efficient tool for the quantitative and qualitative study of many phenomena in different applied areas that otherwise could not have been studied at all. The investigator develops numerical methods that offer some advantages over other techniques for computing solutions of partial differential equations, particularly for problems with complicated geometries or moving boundaries, and applies the methods to a number of illustrative examples.
