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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...