Convergence Analysis of BCF Method for Hookean Dumbbell Model with Finite Difference Scheme

Abstract

Abstract.A convergence analysis of the Brownian configuration fields (BCF) method [M. A. Hulsen, A. P. G. van Heel, and B. H. A. A. van den Brule, J. Non-Newtonian Fluid Mech., 70 (1997), pp. 79-101] for the Hookean dumbbell model with finite difference scheme in dimension 2 or 3 is given in this paper under the assumption that the continuous solution is smooth enough. An explicit solution of the Hookean dumbbell model is obtained via deformation tensor. A large deviation-type estimate for the error of polymeric stress E(QQ) is given, which is a key step in the proof. It is shown that if the number of configuration fields N , the space stepsize h, and the time stepsize δt are chosen appropriately, the convergence of second order in space and first order in time may be proved after excluding a set of small probability. Simultaneous discretization of Monte Carlo and space and the inverse inequality trick are essential for the proof. 1. Introduction. The dumbbell model is the simplest model of polymeric fluids that takes into account the microscopic behavior of the solute polymers [2,4,16]. It models the polymers in dilute solution by dumbbells, each with two beads connected by a spring. The configuration of the spring then specifies the conformation of the polymer. Denote by u and p the velocity and pressure of the fluid and by Q the configuration of the spring; then one has the following coupled macroscopic-kinetic equations:

Mathematical Analysis of Multi-Scale Models of Complex Fluids

“…The convergence analysis of the BCF method is generalized to the high dimensional Hookean dumbbell model in [49]. The basic approach is very similar to the shear flow case presented before.…”

“…The analysis for the Hookean dumbbell model in the shear flow is performed in [17] and [37] independently. It is generalized to the high dimensional Hookean dumbbell model in [49] and the rodlike model in shear flow [48]. The analysis for nonlinear dumbbells is still under investigation.…”

“…The expectation cannot be eliminated, which means a stronger result is needed. The key point in [49] is to prove 33) where is an arbitrarily small positive real number. This goal is achieved by observing that Q is a Gaussian process and the discreteQ n are i.i.d.…”

“…The basic technique [79] is applied in [49] to prove the well-posedness and numerical analysis for a coupled shear flow system. The LCP equation under shear flow environment is:…”

Finite element analysis of a simplified stochastic Hookean dumbbells model arising from viscoelastic flows

“…From the authors knowledge, the only numerical analysis in more than one space dimension is [50]. An implicit finite difference method is considered in the unit square (or cube) with periodic boundary conditions.…”