An automatic approach for the stability analysis of multi-relaxation-time lattice Boltzmann models

Abstract

This paper is concerned with the weighted L2-stability of the lattice Boltzmann method (LBM) for the incompressible Navier-Stokes equations. Such stability is formulated for the linearized LBM and can lead to convergence proof of the nonlinear problem. A key step in this stability analysis is to find an appropriate decomposition for the Jacobian matrix of the collision term. For multi-relaxation-time (MRT) models, it is very difficult to analytically obtain such a decomposition and how to overcome this difficulty for MRT models is still a challenge. In this paper, we propose an automatic approach to prove the weighted L2-stability of the LBM with general collision models. Instead of exploring the decomposition for each specific model, we partially fix the decomposition and turn to show the symmetry and semi-negative definiteness of the other part of the decomposition, which can be automatically examined by a simple computer code. We apply our automatic approach to ten MRT models and stability conditions are generated by the code immediately. In particular, for the popular D2Q9 orthogonal MRT model, the stability condition we obtained has one less constraint than that given in previous work. As verifications, we also provide some analytical proofs for the stability of orthogonal and weighted-orthogonal MRT models. The proofs are much more complicated than the automatic approach, not to mention that those for nonorthogonal models are unavailable. These demonstrate the efficiency and superiority of the proposed automatic approach. With this approach, proving the weighted L2-stability and thereby the convergence of general MRT models becomes straightforward.