In this paper, we develop a simple, efficient, and fifth-order finite difference interpolation-based Hermite WENO (HWENO-I) scheme for one- and two-dimensional hyperbolic conservation laws. We directly interpolate the solution and first-order derivative values and evaluate the numerical fluxes based on these interpolated values. We do not need the split of the flux functions when reconstructing numerical fluxes and there is no need for any additional HWENO interpolation for the modified derivative. The HWENO interpolation only needs to be applied one time which utilizes the same candidate stencils, Hermite interpolation polynomials, and linear/nonlinear weights for the solution and first-order derivative at the cell interface, as well as the modified derivative at the cell center. The HWENO-I scheme inherits the advantages of the finite difference flux-reconstruction-based HWENO-R scheme [Fan et al., Comput. Methods Appl. Mech. Engrg., 2023], including fifth-order accuracy, compact stencils, arbitrary positive linear weights, and high resolution. The HWENO-I scheme is simpler and more efficient than the HWENO-R scheme and the previous finite difference interpolation-based HWENO scheme [Liu and Qiu, J. Sci. Comput., 2016] which needs the split of flux functions for the stability and upwind performance for the high-order derivative terms. Various benchmark numerical examples are presented to demonstrate the accuracy, efficiency, high resolution, and robustness of the proposed HWENO-I scheme.

翻译：本文针对一维和二维双曲守恒律，提出了一种简单、高效且具有五阶精度的基于有限差分插值的Hermite WENO（HWENO-I）格式。我们直接对解函数及其一阶导数值进行插值，并基于这些插值计算数值通量。在重构数值通量时，我们无需对通量函数进行分裂，也无需对修正导数进行任何额外的HWENO插值。HWENO插值仅需应用一次，该过程对单元界面处的解函数、一阶导数以及单元中心处的修正导数，均使用相同的候选模板、Hermite插值多项式以及线性/非线性权值。该HWENO-I格式继承了基于有限差分通量重构的HWENO-R格式[Fan等人，Comput. Methods Appl. Mech. Engrg., 2023]的优点，包括五阶精度、紧致模板、任意的正线性权值以及高分辨率。相较于HWENO-R格式以及先前基于有限差分插值的HWENO格式[Liu和Qiu，J. Sci. Comput., 2016]（后者为保证高阶导数项的稳定性和迎风性能，需要对通量函数进行分裂），本文提出的HWENO-I格式更为简单高效。文中给出了多种基准数值算例，验证了所提HWENO-I格式的精度、效率、高分辨率以及鲁棒性。