For most model equations it is unlikely that exact solutions or convergent series approximations can be found. However, asymptotic expansions can give excellent approximations when there is some small, or large, parameter in the problem. This course introduces the basic ideas and shows how they can be applied to integrals of practical importance that may be difficult to evaluate because of rapidly varying integrands. Then applications to o.d.e.s are covered, with WKB and balance methods. Finally, application to p.d.e.s shows how to approximate eigenvalues and how geometric ray theory can be applied to a range of problems.
Emphasis is put on practical problems that are difficult to solve numerically.
Two assessed projects, submitted at beginning and end of Spring Term. The second project is in combination with the following course. Finite difference, element and volume schemes, Fast Fourier transforms, polynomial representations, accuracy and stability. Iterative solvers, multi-level acceleration, adaptivity. In many industrial systems difficulties arise when small changes in operating conditions lead to large changes in the state of the system.