Computation of highly-oscillatory problems made easy
Abstract
Rapidly oscillating problems, whether differential equations or
integrals, ubiquitous in applications, are allegedly difficult to
compute. In this talk we will endeavour to persuade the audience that
this is false: high oscillation, properly understood, is good for
computation! We describe methods for differential equations, based on
Magnus and Neumann expansions of modified systems, whose efficacy
improves in the presence of high oscillation. Likewise, we analyse
generalised Filon quadrature methods, showing that also their error
sharply decreases as the oscillation becomes more rapid.