Mesoscale Analysis of the Equation-Free Constrained Runs Initialization Scheme

In this article, we analyze the stability, convergence, and accuracy of the constrained runs initialization scheme for a mesoscale lattice Boltzmann model (LBM). This type of initialization scheme was proposed by Gear and Kevrekidis in [J. Sci. Comput., 25 (2005), pp. 17–28] in the context of both singularly perturbed ordinary differential equations and equation-free computing. It maps the given macroscopic initial variables to the higher-dimensional space of microscopic/mesoscopic variables. The scheme performs short runs with the microscopic/mesoscopic simulator and resets the macroscopic variables (typically the lower order moments of the microscopic/mesoscopic variables), while leaving the higher order moments unchanged. We use the LBM Bhatnagar–Gross–Krook (BGK) model for one-dimensional reaction-diffusion systems as the microscopic/mesoscopic model. For such systems, we prove that the constrained runs scheme is unconditionally stable and that it converges to an approximation of the slaved state, i.e., the mesoscopic state which is consistent with the macroscopic initial condition. This approximation is correct up to and including the first order terms in the Chapman–Enskog expansion of the LBM. The asymptotic convergence factor is $|1-\omega|$ with $\omega$ the BGK relaxation parameter. The results are illustrated numerically for the FitzHugh–Nagumo system. Furthermore, we use the constrained runs scheme to perform a coarse equation-free bifurcation analysis of this model. Finally, we show that the constrained runs scheme is very similar to the LBM initialization scheme proposed by Mei et al. in [Comput. Fluids, 35 (2006), pp. 855–862] when implemented for our model problem, and that our numerical analysis applies to the latter scheme also.