A Moment-Based Hermite WENO Scheme with Unified Stencils for Hyperbolic Conservation Laws

In this paper, a fifth-order moment-based Hermite weighted essentially non-oscillatory scheme with unified stencils (termed as HWENO-U) is proposed for hyperbolic conservation laws. The main idea of the HWENO-U scheme is to modify the first-order moment by a HWENO limiter only in the time discretizations using the same information of spatial reconstructions, in which the limiter not only overcomes spurious oscillations well, but also ensures the stability of the fully-discrete scheme proved by the von-Neumann analysis. Benefited by this new framework, the HWENO-U scheme involves only a single HWENO reconstruction throughout the entire spatial discretizations, while previous HWENO schemes have to bring additional procedures. Meanwhile, the HWENO-U scheme can use the artificial linear positive weights (the sum is one), but a normalization is made for the original definition of non-linear weights to achieve scale-invariance, which can reduce problem-specific dependencies especially for simulating the problems with sharp scale variations. Compared with previous HWENO schemes, the HWENO-U scheme is simpler and more efficient for utilizing the same candidate stencils, reconstructed polynomials, and nonlinear weights both in the limiter and the spatial reconstruction. Besides, the HWENO-U scheme has more compact stencils, higher resolutions near discontinuities, and smaller numerical errors in smooth regions than a fifth-order WENO scheme with the same framework. Extensive numerical tests are carried out to validate the efficiency, robustness, accuracy, and resolution of the proposed scheme.