Numerical Analysis

The problem of outer boundary conditions for partial differential equations on unbounded domains has a long history that goes back to Sommerfeld. There have been significant developments in the treatment of artificial outer boundaries, particularly since the 1970s, with the rise and prominence of numerical methods. Such boundary treatments have various names, such as absorbing, complete, radiative, non-reflective, and transparent. We group them into two main categories: local and exact.

Local methods are convenient but approximate; exact methods are accurate but cumbersome.

Geometric intuition and computational experience suggest that hyperboloidal compactification is unique among the existing treatments of the outer boundary problem in the numerical literature: it is local and exact. In addition, the method leads to stable and efficient numerical implementations. This project will demonstrate these properties rigorously and provide guidelines for the optimal choice of free parameters in spacetime transformations.