We introduce Spline Moment Equations (SME) for kinetic equations using a new weighted spline ansatz of the distribution function and investigate the ansatz, the model, and its performance by simulating the one-dimensional Boltzmann-BGK equation. The new basis is composed of weighted constrained splines for the approximation of distribution functions that preserves mass, momentum, and energy. This basis is then used to derive moment equations using a Galerkin approach for a shifted and scaled Boltzmann-BGK equation, to allow for an accurate and efficient discretization in velocity space with an adaptive grid. The equations are given in compact analytical form and we show that the hyperbolicity properties are similar to the well-known Grad moment model. The model is investigated numerically using the shock tube, the symmetric two-beam test and a stationary shock structure test case. All tests reveal the good approximation properties of the new SME model when the parameters of the spline basis functions are chosen properly. The new SME model outperforms existing moment models and results in a smaller error while using a small number of variables for efficient computations.

中文翻译：

---

一维 Boltzmann-BGK 方程的样条矩模型

我们使用分布函数的新加权样条 ansatz 为动力学方程引入样条矩方程 (SME)，并通过模拟一维 Boltzmann-BGK 方程来研究 ansatz、模型及其性能。新的基础由加权约束样条组成，用于近似保留质量、动量和能量的分布函数。然后，该基用于使用伽辽金方法推导出位移和缩放 Boltzmann-BGK 方程的矩方程，以允许在具有自适应网格的速度空间中进行准确有效的离散化。方程以紧凑的解析形式给出，我们表明双曲特性类似于众所周知的 Grad 矩模型。使用激波管对模型进行数值研究，对称两梁试验和静止冲击结构试验箱。当样条基函数的参数选择正确时，所有测试都显示了新 SME 模型的良好近似特性。新的 SME 模型优于现有的矩模型，并在使用少量变量进行高效计算的同时产生更小的误差。