(648e) Stochastic Splitting Methods for Numerical Simulation of Rouse Chains in Flow

Simulation of the Rouse model in flow underlies a great many simulations of polymer dynamics, both in entangled melts and in dilute solution.  The Rouse model in shear flow is interesting for its own sake 
because the resulting chain motion is complex and unsteady.  Individual chains stretch in flow and tumble, when the leading end of the chain dips below the trailing end.  Unsteady motion gives rise to a varied ensemble of molecular configurations, which has been termed ``molecular individualism'' and is most easily explored by direct simulation.

In many applications, considerable CPU time is spent simply evolving the Rouse model forward in time.  Typically, a simple explicit stochastic Euler method is used to evolve the Rouse model.  Here we compare this approach to a stochastic splitting method, which splits the evolution operator into a linear stochastic and nonlinear deterministic part, and exploits an exact formal solution of the linear Rouse model in terms of the noise history.  We show that this splitting method has second-order weak convergence, whereas the stochastic Euler method has only first-order weak convergence.  Most importantly, the splitting method is unconditionally stable, in contrast to the limited stability range of the Euler method.

Analogous splitting methods are applicable to a broad class of problems in stochastic dynamics in which noise competes with ordering and flow to determine steady-state structures.  Examples include mixed binary fluids in flow, block copolymer order-disorder transitions in flow, and flow alignment of smectic or nematic liquid crystals.