  The threshold dynamics algorithm of Merriman, Bence, and Osher is only first
order accurate in the two-phase setting. Its accuracy degrades further to half
order in the multi-phase setting, a shortcoming it has in common with other
related, more recent algorithms such as the equal surface tension version of
the Voronoi implicit interface method. As a first, rigorous step in addressing
this shortcoming, we present two different second order accurate versions of
two-phase threshold dynamics. Unlike in previous efforts in this direction, we
present careful consistency calculations for both of our algorithms. The first
algorithm is consistent with its limit (motion by mean curvature) up to second
order in any space dimension. The second achieves second order accuracy only in
dimension two, but comes with a rigorous stability guarantee (unconditional
energy stability) in any dimension -- a first for high order schemes of its
type.
