We investigate high-order Convolution Quadratures methods for the solution of
the wave equation in unbounded domains in two dimensions that rely on Nystr\"om
discretizations for the solution of the ensemble of associated Laplace domain
modified Helmholtz problems. We consider two classes of CQ discretizations, one
based on linear multistep methods and the other based on Runge-Kutta methods,
in conjunction with Nystr\"om discretizations based on Alpert and QBX
quadratures of Boundary Integral Equation (BIE) formulations of the Laplace
domain Helmholtz problems with complex wavenumbers. We present a variety of
accuracy tests that showcase the high-order in time convergence (up to and
including fifth order) that the Nystr\"om CQ discretizations are capable of
delivering for a variety of two dimensional scatterers and types of boundary
conditions