A Hermite Method with a Discontinuity Sensor for Hamilton–Jacobi Equations

Abstract

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.