Cubature rules for weakly and fully compressible off-lattice Boltzmann methods

TL;DR

This paper demonstrates that using cubature-derived velocity sets in off-lattice Boltzmann methods enhances accuracy and efficiency in simulating compressible flows, outperforming traditional on-lattice velocity sets.

Contribution

It introduces new velocity sets derived from cubature rules for off-lattice Boltzmann methods, improving simulation accuracy and stability for compressible flows.

Findings

Cubature-derived velocity sets can replace traditional Gauss-product rule sets.

Off-lattice D2Q19 velocity set matches the performance of D2Q25 in shock-vortex simulations.

Degree-nine D3Q45 velocity set accurately reproduces kinetic energy in high Mach number flows.

Abstract

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.