Compressible lattice Boltzmann methods with adaptive velocity stencils: An interpolation-free formulation

TL;DR

This paper introduces a new formulation of adaptive lattice Boltzmann methods for compressible flows that eliminates the need for space interpolations, significantly enhancing parallel efficiency and stability at high speeds.

Contribution

It proposes an interpolation-free adaptive LBM formulation that restricts phase discretization to compatible states, enabling on-grid propagation and improved efficiency for high Mach number flows.

Findings

Eliminates space interpolations in adaptive LBMs.

Enables efficient simulation of high-speed compressible flows.

Demonstrates improved parallel efficiency and stability.

Abstract

Adaptive lattice Boltzmann methods (LBMs) are based on velocity discretizations that self-adjust to local macroscopic conditions such as velocity and temperature. While this feature improves the accuracy and the stability of LBMs for large velocity and temperature fluctuations, it also strongly impacts the efficiency of the algorithm due to space interpolations that are required to get populations at grid nodes. To avoid this defect, the present work proposes new formulations of adaptive LBMs for the simulation of compressible flows which do not rely anymore on space interpolations, hence, drastically improving their parallel efficiency for the simulation of high-speed compressible flows. To reach this goal, the adaptive phase discretization is restricted to particular states that are compliant with the efficient "collide and stream" algorithm, and as a consequence it does not require additional interpolation steps. The development of proper state-adaptive solvers with on-grid propagation imposes new restrictions and challenges on the discrete stencils, namely the need for an extended operability range allowing for the transition between two phase discretizations. Achieving the minimum operability range for discrete polynomial equilibria requires rather large stencils (e.g. D2Q81, D2Q121) and is therefore not competitive for compressible flow simulations. However, as shown in the article, the use of numerical equilibria can provide for overlaps in the operability ranges of neighboring discrete shifts at acceptable cost using the D2Q21 lattice. Through several numerical validations, the present approach is shown to allow for an efficient realization of discrete state-adaptive LBMs for high Mach number flows even in the low viscosity regime.