Semi-Lagrangian lattice Boltzmann method for compressible flows

TL;DR

This paper introduces a semi-Lagrangian lattice Boltzmann method for compressible flows that achieves high-order accuracy, supports supersonic regimes, and maintains Galilean invariance through advanced interpolation and grid techniques.

Contribution

The paper develops a novel SLLBM approach with high-order interpolation, static reference frame operation, and flexible velocity sets, improving accuracy and applicability for compressible flow simulations.

Findings

Supports supersonic flows with high Galilean invariance

Achieves high-order spatial accuracy with advanced interpolation

Demonstrates effectiveness on diverse flow problems

Abstract

This work thoroughly investigates a semi-Lagrangian lattice Boltzmann (SLLBM) solver for compressible flows. In contrast to other LBM for compressible flows, the vertices are organized in cells, and interpolation polynomials up to fourth order are used to attain the off-vertex distribution function values. Differing from the recently introduced Particles on Demand (PoD) method, the method operates in a static, non-moving reference frame. Yet the SLLBM in the present formulation grants supersonic flows and exhibits a high degree of Galilean invariance. The SLLBM solver allows for an independent time step size due to the integration along characteristics and for the use of unusual velocity sets, like the D2Q25, which is constructed by the roots of the fifth-order Hermite polynomial. The properties of the present model are shown in diverse example simulations of a two-dimensional Taylor-Green vortex, a Sod shock tube, a two-dimensional Riemann problem and a shock-vortex interaction. It is shown that the cell-based interpolation and the use of Gauss-Lobatto-Chebyshev support points allow for spatially high-order solutions and minimize the mass loss caused by the interpolation. Transformed grids in the shock-vortex interaction show the general applicability to non-uniform grids.