We present a novel positive kinetic scheme built on the efficient collide-and-stream algorithm of the lattice Boltzmann method (LBM) to address hyperbolic conservation laws. We focus on the compressible Euler equations with strong discontinuities. Starting from the work of Jin and Xin [20] and then [4,8], we show how the LBM discretization procedure can yield both first- and second-order schemes, referred to as vectorial LBM. Noticing that the first-order scheme is convex preserving under a specific CFL constraint, we develop a blending strategy that preserves both the conservation and simplicity of the algorithm. This approach employs convex limiters, carefully designed to ensure either positivity (of the density and the internal energy) preservation (PP) or well-defined local maximum principles (LMP), while minimizing numerical dissipation. On challenging test cases involving strong discontinuities and near-vacuum regions, we demonstrate the scheme accuracy, robustness, and ability to capture sharp discontinuities with minimal numerical oscillations.

翻译：我们提出了一种基于格子玻尔兹曼方法高效碰撞-输运算法的新型正性动力学格式，用于求解双曲守恒律。我们重点关注具有强间断的可压缩欧拉方程。从Jin和Xin的工作[20]以及随后的[4,8]出发，我们展示了LBM离散化过程如何导出具有一阶和二阶精度的格式，称为矢量LBM。注意到一阶格式在特定的CFL约束下具有凸保持性，我们发展了一种混合策略，在保持算法守恒性和简洁性的同时，采用精心设计的凸限制器，以确保密度和内能的正性保持或局部极大值原理成立，同时最小化数值耗散。在包含强间断和近真空区域的挑战性测试案例中，我们验证了该格式的精度、鲁棒性以及以最小数值振荡捕捉尖锐间断的能力。