Mathematical Sciences: Numerical Analysis for Time-Dependent Differential Equations

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Stuart The investigator studies the numerical approximation of problems that change in time and are modeled by ordinary or partial differential evolution equations. Two fundamental issues arise in this context. The first is, given a numerical simulation of a problem that evolves in time, how should the data obtained be interpreted: under what circumstances, and in what sense, is the computed data close to the data generated by the underlying differential equation? The first component of the project is concerned with studying this question in the context of the complicated algorithms used by software codes; this is primarily an analytical study requiring an understanding of the discontinuous dynamical systems generated by these algorithms. The second fundamental issue is how to effectively design numerical methods that enable given questions about the equations to be answered efficiently and in a way that yields new insights. The second component of the project is concerned with studying this question in the context of dissipative partial differential equations and, in particular, developing computational techniques for connecting orbits; equations such as the Ginzburg-Landau equation and Kuramoto-Sivashinsky equation are studied. Many problems in science and engineering can be edfectively modeled by means of differential equations of evolution. In practice these equations cannot be solved exactly and computational approximations must be sought. Two important issues arise in this context: (i) ``what is the relationship between computer generated data and the real problem we wish to understand'' and (ii) `` how can efficient algorithms be developed to obtain the type of information that sheds most light on the questions of interest.'' These two issues are addressed in this project in a variety of different contexts. For the first a very broad class of algorithms is studied with potential impact on many areas of science and engineering, including molecular dynamics simulations, planetary interaction simulations and other interesting evolution problems. For the second a number of particular problems are studied that arise from, for example, models of transition to turbulence in fluids and models of complicated combustion processes.