  In this work, we address the imposition of outflow boundary conditions for
one-dimensional conservation laws. While a highly accurate numerical solution
can be obtained in the interior of the domain, boundary discretization can lead
to unphysical reflections. We investigate and implement some classes of
relaxation methods and far-field operators to deal with this problem without
significantly increasing the size of the computational domain. Relations are
established between these techniques, and extensions of them are explored. In
particular, we introduce a simple and robust relaxation method with a
matrix-valued weight function that selectively absorbs outgoing waves. As a
challenging model problem, we consider the Lagrangian formulation of the Euler
equations for an isotropic gas with inflow boundary conditions determined by an
oscillating piston.
