When solving a moving interface problem, the interface can be tracked using a variety of methods. The level set method captures the interface as an isocontour of a scalar level set function. The method has many advantages, including the ability to express many geometric quantities, such as the interface curvature, as derivatives of the level set. However, the numerically constructed level set function may not be smooth enough to compute the required derivatives. Furthermore, the method overall is known for having a high sensitivity to numerical dissipation. The former of these two shortfalls is addressed by augmenting the traditional level set equations with an explicitly tracked interface curvature. The curvature is then updated alongside the level set through an additional advection equation. The latter shortfall is addressed by combining a new velocity extension, that better maintains the signed-distance property of the level set, with a reconstruct-evolve-average approach to advancing the advection equations. The new approach is shown to have less mass loss (numerical dissipation) and better accuracy than comparable level set approaches. Three scenarios are investigated: an interface moving according to an external velocity field, an interface moving according to the interface curvature (mean-curvature flow), and the air-water interface of a water drop moving according to the curvature-dependent fluid velocity (surface-tension driven flow). point on the interface. Thus, the gradient of the SDLS is neither too shallow to see large movements in interface positions from small perturbations in the level set value nor too steep to influence truncation errors when taking derivatives of the level set. In general, there is not a closed-form expression for the SDLS of an arbitrary interface. Thus, it is numerically constructed using a reinitialization process. There are two main variations of reinitialization: PDE-based reinitialization, developed by Peng et al. [21], and the fast marching method (FMM) reinitialization, developed by Sethian [26]. PDE-based reinitialization obtains the SDLS by solving a timedependent PDE, whose steady state solution is the SDLS. In theory, one can get arbitrarily close to the SDLS by evolving the PDE further and further in time. In practice, however, the combination of a smooth signum function and numerical errors from each iteration can cause the zero contour to move during the reinitialization process, leading to errors in the interface location.FMM reinitialization, on the other hand, computes the SDLS by marching signeddistance values out from the interface. This typically involves first using high-order interpolation near the interface to seed the values, then solving the Eikonal equation in a directional fashion. While this method is generally less computationally costly and less prone to movement of the zero contour than that of PDE-based reinitialization, the resulting SDLS is usually less smooth. This can be an issue, for example, if the interface velocity depends on the curva...