IMC Colloquium Series: "From the Unbounded to the (Artificially) Bounded"

Abstract

The study of wave interactions with obstacles has a rich mathematical history. Interesting and challenging issues arise when we numerically simulate these phenomena. For example, we can only discretize a bounded region, so one method is to introduce an artificial surface enclosing the obstacle. Instead of allowing waves to scatter off bounded obstacles in free space, we now have to study their action inside this computational box. What effect does the truncation have on the original model problem, which is posed in all of space? What further effects do discretization errors bring? We quickly survey some techniques for describing the so-called non-reflecting boundary conditions, and present some new work on the construction of exact Dirichlet-to-Neumann maps. Along the way, we need tools from functional analysis, PDE theory, numerical analysis and asymptotics.