A Duhamel approach for the Langevin equations with holonomic constraints

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. To simulate polymer flows in microscale environments we have developed a numerical method that couples stochastic particle dynamics with an efficient incompressible Navier – Stokes solver. Here, we examine properties of the particle solver alone. We derive a Duhamel-form stochastic particle method for freely jointed polymers and demonstrate that it achieves 2-order weak convergence and 3/2-order strong convergence with holonomic constraints. For time steps approaching the 1=g relaxation time, our method displays greatly enhanced stability relative to comparable solvers based on linearised dynamics. Under these same conditions, our method has solution errors that are approximately six orders of magnitude smaller than that for the linearised algorithm. 1. Introduction The dynamics of a continuum fluid with discrete embedded polymers is important for certain microfluidic applications (e.g. so-called lab-on-a-chip devices used for biochemical analysis and detection) and for modelling viscoelastic phenomena in the dilute limit. Towards this end, we proposed a fluid–particle coupling strategy [12] that uses Brownian dynamics to approximate molecular-level fluid– polymer interactions. In subsequent work (e.g. [7]) the time stability of the scheme was improved, and constraints such as the non-crossing constraint for polymer – polymer interaction were considered. In this short paper, we address the accuracy of our scheme. We work here in the framework of a freely-jointed chain (no polymer–polymer interactions), we consider the fluid velocity field to be prescribed, and we do not consider any rigid domain boundaries. In the context of rigid constraint dynamics (vs. soft penalty method constraints), these omitted interactions will diminish the order of the local discretisation error. Recently, [13] proposed a weak second-order stochas-tic particle dynamics approach that is broadly similar to ours as described in [7,12]. Our approach differs from theirs in our handling of …