We propose a new second-order accurate lattice Boltzmann formulation for linear elastodynamics that is stable for arbitrary combinations of material parameters under a CFL-like condition. The construction of the numerical scheme uses an equivalent first-order hyperbolic system of equations as an intermediate step, for which a vectorial lattice Boltzmann formulation is introduced. The only difference to conventional lattice Boltzmann formulations is the usage of vector-valued populations, so that all computational benefits of the algorithm are preserved. Using the asymptotic expansion technique and the notion of pre-stability structures we further establish second-order consistency as well as analytical stability estimates. Lastly, we introduce a second-order consistent initialization of the populations as well as a boundary formulation for Dirichlet boundary conditions on 2D rectangular domains. All theoretical derivations are numerically verified by convergence studies using manufactured solutions and long-term stability tests.

翻译：我们提出了一种新的二阶精确格子玻尔兹曼格式，用于求解线性弹性动力学问题。该格式在满足类CFL条件下，对于任意的材料参数组合均保持稳定。数值格式的构建采用了一个等价的一阶双曲型方程组作为中间步骤，并为此引入了一个矢量格子玻尔兹曼公式。与传统格子玻尔兹曼公式的唯一区别在于使用了矢量值分布函数，从而保留了算法的所有计算优势。通过渐近展开技术和预稳定性结构的概念，我们进一步建立了格式的二阶一致性以及解析的稳定性估计。最后，我们提出了分布函数的二阶一致初始化方法，以及针对二维矩形区域上Dirichlet边界条件的边界处理公式。所有理论推导均通过使用构造解的收敛性研究和长期稳定性测试进行了数值验证。