High-order central Hermite WENO schemes on staggered meshes for hyperbolic conservation laws

Abstract

In this paper, we propose a class of high-order schemes for solving one- and two-dimensional hyperbolic conservation laws. The methods are formulated in a central finite volume framework on staggered meshes, and they involve Hermite WENO (HWENO) reconstructions in space, and Lax–Wendroff type discretizations or the natural continuous extension of Runge–Kutta methods in time. Compared with central WENO methods, the spatial reconstruction used here is much more compact; and unlike the original HWENO methods, our proposed schemes require neither flux splitting nor the use of numerical fluxes. In the system case, local characteristic decomposition is applied in the reconstructions of cell averages to enhance the non-oscillatory property of the methods. The high resolution and robustness of the methods in capturing smooth and non-smooth solutions are demonstrated through a collection of one- and two-dimensional scalar and system of examples.