The efficiency and robustness of various numerical schemes have been evaluated by performing Brownian dynamics simulations of bead-rod and three popular nonlinear bead-spring chain models in uniaxial extension and simple shear flow. The bead-spring models include finitely extensible nonlinear elastic (FENE) springs, worm-like chain (WLC) springs, and Pade approximation to the inverse Langevin function (ILC) springs. For the bead-spring chains two new predictor-corrector algorithms are proposed, which are much superior to commonly used explicit and other fully implicit schemes. In the case of bead-rod chain models, the mid-point algorithm of Liu [J. Chem. Phys. 90 (1989) 5826] is found to be computationally more efficient than a fully implicit Newton's method. Furthermore, the accuracy and computational efficiency of two different stress expressions for the bead-rod chains, namely the Kramers-Kirkwood and the modified Giesekus have been evaluated under both transient and steady conditions. It is demonstrated that the Kramers-Kirkwood with stochastic filtering is the preferred choice for transient flow while the Giesekus expression is better suited for steady state calculations. The issue of coarse graining from a bead-rod chain to a bead-spring chain has also been investigated. Though bead-spring chains are shown to capture only semi-quantitatively the response of the bead-rod chains in transient flows, a systematic coarse-graining procedure that provides the best description of bead-rod chains via bead-spring chains is presented.
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