High-order moment-based Hermite WENO schemes for hyperbolic conservation laws on triangular meshes

Abstract

In this paper, we construct high-order Hermite weighted essentially non-oscillatory (HWENO) schemes for two-dimensional hyperbolic conservation laws on triangular meshes. These schemes integrate both zeroth- and first-order moments into spatial discretizations, yielding more compact stencils than same-order WENO schemes. Specifically, the third- and fifth-order HWENO schemes require only one and two layers of stencils, respectively, as opposed to the two layers needed by a third-order WENO scheme. Meanwhile, the HWENO schemes demonstrate reduced numerical errors in smooth areas and improved resolutions near discontinuities. Although the HWENO schemes include two auxiliary equations, they retain a unified nonlinear reconstruction process similar to that of WENO schemes. This design choice leads to a modest increase in computational expense and algorithm complexity. Crucially, an efficient definition of smoothness indicators is introduced, based on a midpoint numerical integration of the original definition. This streamlined definition enhances computational efficiencies on unstructured meshes and results in only minor variations in smoothness measurement between the two definitions, regardless of whether the problem is smooth or discontinuous. The HWENO schemes are distinguished by their strong practicality on triangular meshes, with efficient computation of smoothness indicators, consistent use of a single set of compact stencils, and application of artificial linear weights. Extensive numerical experiments are conducted to verify the high-order accuracies, efficiencies, resolutions, robustness, scale-invariance, and the effectiveness of the smoothness indicator for the proposed HWENO schemes.