Abstract
In this paper, a new type of high-order finite volume and finite difference multi-resolution Hermite weighted essentially non-oscillatory (HWENO) schemes are designed for solving hyperbolic conservation laws on structured meshes. Here we only use the information defined on a hierarchy of nested central spatial stencils but do not introduce any equivalent multi-resolution representation, the terminology of multi-resolution HWENO follows that of the multi-resolution WENO schemes (Zhu and Shu, 2018) [29]. The main idea of our spatial reconstruction is derived from the original HWENO schemes (Qiu and Shu, 2004) [19], in which both the function and its first-order derivative values are evolved in time and used in the reconstruction. Our HWENO schemes use the same large stencils as the classical HWENO schemes which are narrower than the stencils of the classical WENO schemes for the same order of accuracy. Only the function values need to be reconstructed by our HWENO schemes, the first-order derivative values are obtained from the high-order linear polynomials directly. Furthermore, the linear weights of such HWENO schemes can be any positive numbers as long as their sum equals one, and there is no need to do any modification or positivity-preserving flux limiting in our numerical experiments. Extensive benchmark examples are performed to illustrate the robustness and good performance of such finite volume and finite difference HWENO schemes.
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