In this paper we present and utilize a new theoretical and computational framework for computing solutions of higher classes of one dimensional transient Navier-Stokes partial differential equations in Lagrangian frame of reference using ρ, u, T variables. The approach utilizes ‘strong form’ of the governing differential equations (GDEs) and least squares method in constructing integral form. The currently used finite element approaches seek convergence of a solution in a fixed order space by h, p, or hp-adaptive processes. The fundamental point of departure in the proposed approach is that we seek the convergence of the computed converged solution over the spaces of different orders containing the basis functions. With this approach, dramatically higher convergence rates than those obtained for h, or hp-processes are achievable and the sequence of progressively converged solutions over the spaces of progressively increasing order in fact converge to the strong solution (analytical or theoretical) of the partial differential equation. It is demonstrated using one-dimensional transient Navier-Stokes equations for compressible fluid flow in Lagrangian frame of reference in ρ, u, T that in the presence of physical diffusion and dissipation, our computed converged solutions have exactly the same characteristics as the strong solutions. Compression of air in a rigid cylinder is used as the model problem.