A numerical sufficiency test for the asymptotic stability of linear time-varying Hurwitz systems is proposed. The algorithmic procedure constructs a bounding tube in which the state is guaranteed to stay. The continuous-time system is evaluated at discrete time instants, for which successive quadratic Lyapunov functions are generated. The tube is constructed based on: (i) a conservative estimate of the state evolution, from a discrete time instant to the next, obtained from the corresponding Lyapunov function, and (ii) re-evaluation of the tube diameter at each discrete time instant to account for variations in the plant matrix. The numerical test is illustrated in simulation via both a stable and an unstable system.