  Computing the non-equilibrium dynamics that follows a quantum quench is
difficult, even in exactly solvable models. Results are often predicated on the
ability to compute overlaps between the initial state and eigenstates of the
Hamiltonian that governs time evolution. Except for a handful of known cases,
it is generically not possible to find these overlaps analytically. Here we
develop a numerical approach to preferentially generate the states with high
overlaps for a quantum quench starting from the ground state or an excited
state of an initial Hamiltonian. We use these preferentially generated states,
in combination with a "high overlap states truncation scheme" and a
modification of the numerical renormalization group, to compute non-equilibrium
dynamics following a quench in the Lieb-Liniger model. The method is
non-perturbative, works for reasonable numbers of particles, and applies to
both continuum and lattice systems. It can also be easily extended to more
complicated scenarios, including those with integrability breaking.
