Off-lattice Boltzmann methods increase the flexibility and applicability of lattice Boltzmann methods by decoupling the discretizations of time, space, and particle velocities. However, the velocity sets that are mostly used in off-lattice Boltzmann simulations were originally tailored to on-lattice Boltzmann methods. In this contribution, we show how the accuracy and efficiency of weakly and fully compressible semi-Lagrangian off-lattice Boltzmann simulations are increased by velocity sets derived from cubature rules, i.e., multivariate quadratures not produced by the Gauß-product rule. In particular, simulations of 2D shock-vortex interactions indicate that the cubature-derived degree-nine D2Q19 velocity set is capable of replacing the Gauß-product rule-derived D2Q25. Likewise, the degree-five velocity sets D3Q13 and D3Q21, as well as a degree-seven D3V27 velocity set were successfully tested for 3D Taylor–Green vortex flows to challenge and surpass the quality of the customary D3Q27 velocity set. In compressible 3D Taylor–Green vortex flows with Mach numbers $\mathrm{Ma}=\{0.5;1.0;1.5;2.0\}$ on-lattice simulations with velocity sets D3Q103 and D3V107 showed only limited stability, while the off-lattice degree-nine D3Q45 velocity set accurately reproduced the kinetic energy provided by literature.

离格Boltzmann方法通过将时间，空间和粒子速度的离散化解耦，从而提高了格子Boltzmann方法的灵活性和适用性。但是，通常在非晶格Boltzmann模拟中使用的速度集最初是针对晶格Boltzmann方法量身定制的。在此贡献中，我们展示了如何通过由库氏规则（即，不是由Gauß乘积规则产生的多元正交）得出的速度集来提高弱且完全可压缩的半拉格朗日离格子Boltzmann模拟的准确性和效率。特别是，二维冲击-涡流相互作用的仿真表明，以孵化器为基础的九度D2Q19速度集能够替代以Gauß-乘积法则为基础的D2Q25。同样，五度速度组D3Q13和D3Q21 以及针对7D D3V27速度集的3D Taylor-Green涡流测试成功，从而挑战并超越了常规D3Q27速度集的质量。在可压缩的3D Taylor-Green涡流中，马赫数$\mathrm{嘛}=\{0。5;\mathrm{1个}。0;\mathrm{1个}。5;\mathrm{2个}。0\}$ 带有速度集D3Q103和D3V107的晶格上模拟仅显示出有限的稳定性，而离格度为9的D3Q45速度集准确地再现了文献中提供的动能。