Fifth-order finite volume derivative-based Hermite WENO scheme with unified stencils for hyperbolic conservation laws

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Original abstract

In this paper, we propose a derivative-based finite volume Hermite WENO (HWENO) scheme for hyperbolic conservation laws, where both the solution and its first-order derivatives are evolved in time and utilized in spatial reconstructions. The key challenge for solving hyperbolic conservation laws is the possible emergence of discontinuities in the numerical solutions. When facing discontinuities, the derivatives can become excessively large, which may compromise the robustness of HWENO schemes. In the first HWENO scheme, different sets of stencils were adopted for reconstructing the governing equation and the derivative equation, respectively, aiming to reduce the influence of the derivatives while preserving high-order accuracy. However, this approach not only substantially increases computational cost but also introduces considerable algorithmic complexity. To overcome these limitations, we exclude the information of the target cell's derivatives from spatial reconstructions, while employing the same reconstructed polynomial during temporal evolution to limit the derivatives. This strategy enhances the robustness of traditional HWENO schemes and allows unified stencils within the derivative-based HWENO framework. Furthermore, the proposed scheme supports arbitrary positive linear weights that sum to one and maintains a compact stencil. Numerical results demonstrate the high-order accuracy, efficiency, high resolution, and robustness of the proposed HWENO scheme.

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• Official / arXiv page
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