Society's ever-increasing integration of autonomous systems in day-to-day life has simultaneously brought forth concerns as to how their safety and reliability can be verified. To this end, reachable sets lend themselves well to this task. These sets describe collections of states that a dynamical system can reach in finite time, which can be used to guarantee goal satisfaction in controller design or to verify that unsafe regions will be avoided. However, general-purpose methods for computing these sets suffer from the curse-of-dimensionality, which typically prohibits their use for systems with more than a small number of states, even if they are linear. In this talk, we derive dynamics for a union and intersection of ellipsoidal sets that, respectively, under-approximate and over-approximate the reachable set for linear time-varying systems subject to an ellipsoidal input constraint and an ellipsoidal terminal (or initial) set. This result arises from the construction of a local viscosity supersolution and subsolution of a Hamilton-Jacobi-Bellman equation for the corresponding reachability problem. The proposed ellipsoidal sets can be generated with polynomial computational complexity in the number of states, making the approximation scheme computationally tractable for continuous-time linear time-varying systems of relatively high dimension.