Mathematical Sciences: Analytical and Numerical Studies of Nonlinear Hyperbolic Systems

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This project involves the study of nonlinear hyperbolic systems of partial differential equations from both analytical and numerical points of view, with emphasis on problems from Plasma Physics and the Vlasov-Maxwell system. Particle Methods for higher-dimensional systems will be studied, with the goal of obtaining convergence as well as higher-order accuracy. Collisional effects (e.g. those from the LANDAU system) will also be included. Many of the presently-used numerical schemes have little (if any) theoretical justification. For those few that are well founded, the known accuracy is relatively poor. Thus from a practical point of view, much improvement is possible. Moreover, at this time there are no rigorous studies (neither theoretical nor computational) of collisional effects. These studies will have application in areas such as fusion energy.