Implicit micro-macro methods

Unlike in continuum approaches, time-dependent micro--macro calculations have been performed up to now almost exclusively using a simple explicit time marching algorithm, the exception being the pioneering work by Somasi and Khomami [J. Non-Newtonian Fluid Mech. 93 (2000) 339] whose iteration of a semi-implicit algorithm is equivalent to a fully implicit formulation. The use of explicit time integration puts explicit micro-macro methods at a disadvantage, since they lack the desirable stability of fully implicit methods. The limitation of using explicit time marching schemes often leads to reduced computational efficiency, since unnecessarily small time steps must be taken to ensure numerical stability. It is shown in a rather general way that micro-macro methods can indeed be treated in a fully implicit fashion, thus putting them on the same footing, as far as stability of time integration is concerned, as continuum approaches. (C) 2004 Published by Elsevier B.V.