Fifth-Order Weighted Power-ENO Schemes for Hamilton-Jacobi Equations

We design a class of Weighted Power-ENO (Essentially Non-Oscillatory) schemes to approximate the viscosity solutions of Hamilton-Jacobi (HJ) equations. The essential idea of the Power-ENO scheme is to use a class of extended limiters to replace the minmod type limiters in the classical third-order ENO schemes so as to improve resolution near kinks where the solution has discontinuous gradients. Then a weighting strategy based on appropriate smoothness indicators lifts the scheme to be fifth-order accurate. In particular, numerical examples indicate that the Weighted Power3ENO5 works for general HJ equations while the Weighted Power∞ENO5 works for non-linear convex HJ equations. Numerical experiments also demonstrate the accuracy and the robustness of the new schemes.