  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 is increased by velocity sets derived from cubature
rules, i.e. multivariate quadratures, which have not been 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 to
replace 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 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.
