Mathematical Sciences: The Numerical Analysis of Evolution Equations Over Long Time Intervals

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The investigator will analyse and design numerical methods for initial value problems whose solutions are required over long time intervals. Two classes of equations will be studied: those possessing an absorbing set, where the long-time dynamics may be extremely complex (chaotic) but remain in a bounded set; and with those a gradient structure, so that solutions approach the set of (typically multiple) steady states for large time. For fixed time-stepping strategies, the proposed area of research is a generalisation of the classical theories of A, B and G stability. These classical theories apply to classes of problems with simple dynamics and it is of interest to determine how the theories can be used and modified to cope with more complicated dynamical problems. For variable time-stepping strategies some existing work on linear problems will be generalised to nonlinear problems. Work will also be directed towards the approximation of homoclinic and heteroclinic orbits, which play an important role in the long-time dynamics of the two classes of equation under consideration. Many phenomena of interest in the physical sciences and engineering require the understanding of dynamical phenomena that evolve over very long time scales. Examples include turbulence in fluids, phase separation in solids, and planetary motions. Often these phenomena are modelled by differential equations and it is necessary to approximate these equations by computational techniques to obtain information about the underlying problem being modelled. A question of fundamental importance is to ascertain the validity of the computer-generated output and, in particular, it is important to understand fully the relationship between the computer-generated approximations of the model and the behavior of the model itself. The study of this problem is well understood in many simple situations but these do not include situations where long time scales are present. The object of the research is to study this problem, using the underlying theory of the differential equation models to guide the study of the computer-generated approximations.