High-order semi-discrete central-upwind schemes for multi-dimensional Hamilton–Jacobi equations

We present the first fifth-order, semi-discrete central-upwind method for approximating solutions of multi-dimensional Hamilton–Jacobi equations. Unlike most of the commonly used high-order upwind schemes, our scheme is formulated as a Godunov-type scheme. The scheme is based on the fluxes of Kurganov–Tadmor and Kurganov–Noelle–Petrova, and is derived for an arbitrary number of space dimensions. A theorem establishing the monotonicity of these fluxes is provided. The spatial discretization is based on a weighted essentially non-oscillatory reconstruction of the derivative. The accuracy and stability properties of our scheme are demonstrated in a variety of examples. A comparison between our method and other fifth-order schemes for Hamilton–Jacobi equations shows that our method exhibits smaller errors without any increase in the complexity of the computations.