Topics in matrix analysis
Stability analysis of lattice Boltzmann methods
Journal of Computational Physics
A Stability Notion for Lattice Boltzmann Equations
SIAM Journal on Scientific Computing
Asymptotic analysis of the lattice Boltzmann equation
Journal of Computational Physics
Convergence of lattice Boltzmann methods for Stokes flows in periodic and bounded domains
Computers & Mathematics with Applications
Weighted $\mathbb{L}^2$-Stability of the Lattice Boltzmann Method
SIAM Journal on Numerical Analysis
Editorial: Mesoscopic methods in engineering and science
Computers & Mathematics with Applications
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The stability structure for lattice Boltzmann schemes has been introduced in Banda et al. (2006) [16], Junk and Yong (2007) [14] to analyze the stability of numerical algorithms. The first purpose of this paper is to discuss the stability structure from the perspective of matrix analysis. Its second goal is to illustrate and apply the results to different classes of lattice Boltzmann collision operators. In particular we formulate an equivalence condition-just recently also reported in Yong (2008) [18]-that guarantees the existence of a pre-stability structure. It is then illustrated by several examples, how this equivalence condition can be effectively employed for the systematic verification and construction of stable collision operators. Finally, we point out some shortcomings of the stability structure approach arising in certain cases.