In this paper, we are concerned about the lattice Boltzmann methods (LBMs) based on vector-kinetic models for hyperbolic partial differential equations. In addition to usual lattice Boltzmann equation (LBE) derived by explicit discretisation of vector-kinetic equation (VKE), we also consider LBE derived by semi-implicit discretisation of VKE and compare the relaxation factors of both. We study the properties such as H-inequality, total variation boundedness and positivity of both the LBEs, and infer that the LBE due to semi-implicit discretisation naturally satisfies all the properties while the LBE due to explicit discretisation requires more restrictive condition on relaxation factor compared to the usual condition obtained from Chapman-Enskog expansion. We also derive the macroscopic finite difference form of the LBEs, and utilise it to establish the consistency of LBEs with the hyperbolic system. Further, we extend this LBM framework to hyperbolic conservation laws with source terms, such that there is no spurious numerical convection due to imbalance between convection and source terms. We also present a D$2$Q$9$ model that allows upwinding even along diagonal directions in addition to the usual upwinding along coordinate directions. The different aspects of the results are validated numerically on standard benchmark problems.

翻译：本文研究了基于向量-动力学模型的双曲型偏微分方程的格子Boltzmann方法（LBMs）。除通过对向量-动力学方程（VKE）进行显式离散化得到常规格子Boltzmann方程（LBE）外，我们还考虑了通过半隐式离散化VKE得到的LBE，并对两者的松弛因子进行了比较。我们研究了两种LBE的H-不等式、总变差有界性和正性等性质，发现半隐式离散化得到的LBE自然满足所有性质，而显式离散化得到的LBE对松弛因子的限制条件比Chapman-Enskog展开得到的常规条件更为严格。我们还推导了LBE的宏观有限差分形式，并利用该形式建立了LBE与双曲型系统的一致性。进一步地，我们将该LBM框架推广至带源项的双曲守恒律，使得对流项与源项之间的不平衡不会产生虚假数值对流。此外，我们提出了一个D$2$Q$9$模型，该模型除沿坐标方向的常规迎风外，还允许沿对角线方向进行迎风处理。通过标准基准算例对上述结果的各个方面进行了数值验证。