The viscous stagnation-flow solidification problem for a pure substance is investigated by means of quasi-steady, instantaneous-similarity, as well as finite-difference methods. The liquid velocity field, the solid and liquid temperature distributions, the solid-liquid interface location, as well as its growth rate, are obtained using all three methods and comparisons of the solutions are made. The instantaneous-similarity solution at a sufficiently small time is used as the initial field to start the finite-difference calculation. All three methods show the existence of an asymptotic limit of the solidification-front position. The effect on this limit of several important dimensionless parameters, including the Prandtl number, is presented.