  Including derivative information in the modelling of moving interfaces has
been proposed as one method to increase the accuracy of numerical schemes with
minimal additional cost. Here a new level set reinitialization technique using
the fast marching method is presented. This augmented fast marching method will
calculate the signed distance function and up to the second-order derivatives
of the signed distance function for arbitrary interfaces. In addition to
enforcing the condition $|\nabla\phi|^2=1$, where $\phi$ is the level set
function, the method ensures that $\nabla(|\nabla\phi|)^2=0$ and
$\nabla\nabla(|\nabla\phi|)^2=0$ are also satisfied. Results indicate that for
both two- and three-dimensional interfaces the resulting level set and
curvature field are smooth even for coarse grids. Convergence results show that
using first-order upwind derivatives and the augmented fast marching method
result in a second-order accurate level set and gradient field and a
first-order accurate curvature field.
