  We present a Hermite interpolation based partial differential equation solver
for Hamilton-Jacobi equations. Many Hamilton-Jacobi equations have a nonlinear
dependency on the gradient, which gives rise to discontinuities in the
derivatives of the solution, resulting in kinks. We built our solver with two
goals in mind: 1) high order accuracy in smooth regions and 2) sharp resolution
of kinks. To achieve this, we use Hermite interpolation with a smoothness
sensor. The degrees-of freedom of Hermite methods are tensor-product Taylor
polynomials of degree $m$ in each coordinate direction. The method uses
$(m+1)^d$ degrees of freedom per node in $d$-dimensions and achieves an order
of accuracy $(2m+1)$ when the solution is smooth. To obtain sharp resolution of
kinks, we sense the smoothness of the solution on each cell at each timestep.
If the solution is smooth, we march the interpolant forward in time with no
modifications. When our method encounters a cell over which the solution is not
smooth, it introduces artificial viscosity locally while proceeding normally in
smooth regions. We show through numerical experiments that the solver sharply
captures kinks once the solution losses continuity in the derivative while
achieving $2m+1$ order accuracy in smooth regions.
