Lifting in equation-free methods for molecular dynamics simulations of dense fluids

Abstract

Within the context of multiscale computations, equation-free methodshave been developed. In this approach, the evolution of a system issimulated on the macroscopic level while only a microscopic model isexplicitly available. To this end, a coarse time stepper forthe macroscopic variables can be constructed, based on appropriatelyinitialized microscopic simulations. In this paper, we investigatethe initialization of the microscopic simulator using the macroscopicvariables only (called lifting in the equation-free framework)when the microscopic model is a molecular dynamics (MD) description ofa mono-atomic dense fluid. We assume a macroscopic model to exist interms of the lowest order velocity moments of the particledistribution (density, velocity and temperature). The major difficultyis to design a lifting operator that accurately reconstructs thephysically correct state of the fluid (i.e., the higher order moments)at a reasonable computational cost. We construct a lifting operator,as well as a restriction operator for the reverse mapping.For a simple model problem, we perform asystematic numerical study to assess the time scales on which thelifting errors disappear after reinitialization (healing); wealso examine the effects on the simulated macroscopic behavior.Theresults show that, although in some cases accurate initialization ofthe higher order moments is not crucial, in general a detailed studyof the lifting operator is required.