High-Order Multi-resolution Central Hermite WENO Schemes for Hyperbolic Conservation Laws

In this paper, a class of high-order multi-resolution central Hermite WENO (C-HWENO) schemes for solving hyperbolic conservation laws is proposed. Formulated in a central finite volume framework on staggered meshes, the methods adopt the multi-resolution HWENO reconstructions (Li et al. in J Comput Phys 446:110653, 2021; Li et al. in Commun Comput Phys 32(2): 364-400, 2022) in space and the natural continuous extension of Runge–Kutta methods in time. Based on the zeroth-order and first-order moments of the solution defined on a series of hierarchical central spatial stencils, the proposed methods are sixth-order while the C-HWENO methods by Tao et al. (J Comput Phys 318:222-251, 2016) are fifth-order in accuracy. The linear weights of such HWENO reconstructions can be any positive numbers as long as their sum equals one, which leads to much simpler implementation and better cost efficiency than the methods by Tao et al. (J Comput Phys 318:222-251, 2016). The first-order moments are modified and the HWENO reconstructions are applied in the troubled-cells, while the linear reconstructions are used for the rest. Meanwhile, our new methods have compact stencils in the reconstructions and require neither numerical fluxes nor flux splitting. Extensive one- and two-dimensional numerical examples are performed to illustrate the accuracy and high resolution of the new C-HWENO schemes.