An oscillation-free Hermite WENO scheme for hyperbolic conservation laws

• Published in: Science China. Mathematics Vol. 67; no. 2; pp. 431 - 454
• Main Authors: Zhao, Zhuang, Qiu, Jianxian
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01.02.2024; Springer Nature B.V
• Subjects: Applications of Mathematics; Conservation laws; Damping; Mathematics; Mathematics and Statistics; Polynomials; Reconstruction; Robustness (mathematics); hyperbolic conservation laws; discontinuous Galerkin method; 65M08; Hermite WENO scheme; oscillation-free; 35L65; adaptive order
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-022-2064-1

In this paper, the sixth-order oscillation-free Hermite weighted essentially non-oscillatory (OF-HWENO) scheme is proposed for hyperbolic conservation laws on structured meshes, where the zeroth- and first-order moments are the variables for the governing equations. The main difference from other HWENO schemes existed in the literature is that we add high-order numerical damping terms in the first-order moment equations to control spurious oscillations for the OF-HWENO scheme. The OF-HWENO scheme not only can achieve the designed optimal numerical order, but also can be easily implemented as we use only one set of stencil in the reconstruction procedure and the same reconstructed polynomials are applied for the zeroth- and first-order moments equations. In order to obtain the adaptive order resolution when facing the discontinuities, a transition polynomial is added in the reconstruction, where the associated linear weights can also be any positive numbers as long as their summation equals one. In addition, the OF-HWENO scheme still keeps the compactness as only immediate neighbor values are needed in the space discretization. Some benchmark numerical tests are performed to illustrate the high-order accuracy, high resolution and robustness of the proposed scheme.