An $h$-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations

In [35, 36], we presented an $h$-adaptive Runge-Kuttadiscontinuous Galerkin method using troubled-cell indicators forsolving hyperbolic conservation laws. A tree data structure (binarytree in one dimension and quadtree in two dimensions) is used to aidstorage and neighbor finding. Mesh adaptation is achieved byrefining the troubled cells and coarsening the untroubled"children". Extensive numerical tests indicate that the proposed $h$-adaptive method is capable of saving the computational cost andenhancing the resolution near the discontinuities. In this paper, weapply this $h$-adaptive method to solve Hamilton-Jacobi equations,with an objective of enhancing the resolution near thediscontinuities of the solution derivatives. One- andtwo-dimensional numerical examples are shown to illustrate thecapability of the method.