  The level set approach represents surfaces implicitly, and advects them by
evolving a level set function, which is numerically defined on an Eulerian
grid. Here we present an approach that augments the level set function values
by gradient information, and evolves both quantities in a fully coupled
fashion. This maintains the coherence between function values and derivatives,
while exploiting the extra information carried by the derivatives. The method
is of comparable quality to WENO schemes, but with optimally local stencils
(performing updates in time by using information from only a single adjacent
grid cell). In addition, structures smaller than the grid size can be located
and tracked, and the extra derivative information can be employed to obtain
simple and accurate approximations to the curvature. We analyze the accuracy
and the stability of the new scheme, and perform benchmark tests.
