Numerical simulations on infinite domains are challenging. In this talk, we will take geometric approaches to analyze the problems and provide new solutions.  One problem we tackle is the perfectly matched layer (PML) problem for computational waves on infinite domains. PML is a theoretical wave-absorbing medium attached to the truncated domain that generates no reflection at the interface. However, previous methods suffer from numerical reflections due to discretization error. We derive the PML based on principles in discrete differential geometry, and for the first time, we obtain a discrete PML that generates no numerical reflections.