  This paper is concerned with the development of suitable numerical method for
the approximation of discontinuous solutions of parameter-dependent linear
hyperbolic conservation laws. The objective is to reconstruct such
approximation, for new instances of the parameter values for any time, from a
transformation of pre-computed snapshots of the solution trajectories for new
parameter values. In a finite volume setting, a Reconstruct-Transform-Average
(RTA) algorithm inspired from the Reconstruct-Evolve-Average one of Godunov's
method is proposed. It allows to perform, in three steps, a transformation of
the snapshots with piecewise constant reconstruction. The method is fully
detailed and analyzed for solving a parameter-dependent transport equation for
which the spatial transformation is related to the characteristic intrinsic to
the problem. Numerical results for transport equation and linear elastodynamics
equations illustrate the good behavior of the proposed approach.
