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Abstract
Within the context of multiscale computations, equation-free methods
have been developed. In this approach, the evolution of a system is
simulated on the macroscopic level while only a microscopic model is
explicitly available. To this end, a coarse time stepper for
the macroscopic variables can be constructed, based on appropriately
initialized microscopic simulations. In this paper, we investigate
the initialization of the microscopic simulator using the macroscopic
variables only (called lifting in the equation-free framework)
when the microscopic model is a molecular dynamics (MD) description of
a mono-atomic dense fluid. We assume a macroscopic model to exist in
terms of the lowest order velocity moments of the particle
distribution (density, velocity and temperature). The major difficulty
is to design a lifting operator that accurately reconstructs the
physically 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 a
systematic numerical study to assess the time scales on which the
lifting errors disappear after reinitialization (healing); we
also examine the effects on the simulated macroscopic behavior.
The
results show that, although in some cases accurate initialization of
the higher order moments is not crucial, in general a detailed study
of the lifting operator is required.
Mathematics Subject Classification: Primary: 70G60.
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