Adaptive Central-Upwind Schemes for Hamilton–Jacobi Equations with Nonconvex Hamiltonians

Abstract

This paper is concerned with computing viscosity solutions of Hamilton---Jacobi equations using high-order Godunov-type projection-evolution methods. These schemes employ piecewise polynomial reconstructions, and it is a well-known fact that the use of more compressive limiters or higher-order polynomial pieces at the reconstruction step typically provides sharper resolution. We have observed, however, that in the case of nonconvex Hamiltonians, such reconstructions may lead to numerical approximations that converge to generalized solutions, different from the viscosity solution. In order to avoid this, we propose a simple adaptive strategy that allows to compute the unique viscosity solution with high resolution. The strategy is not tight to a particular numerical scheme. It is based on the idea that a more dissipative second-order reconstruction should be used near points where the Hamiltonian changes convexity (in order to guarantee convergence to the viscosity solution), while a higher order (more compressive) reconstruction may be used in the rest of the computational domain in order to provide a sharper resolution of the computed solution. We illustrate our adaptive strategy using a Godunov-type central-upwind scheme, the second-order generalized minmod and the fifth-order weighted essentially non-oscillatory (WENO) reconstruction. Our numerical examples demonstrate the robustness, reliability, and non-oscillatory nature of the proposed adaptive method.