Approaches for Fast Brownian Dynamics Simulation with Constraints

Abstract

Constraints can be used to eliminate quickly fluctuating degrees of freedom in dynamical systems enabling solutions that resolve the behavior of the system with fewer time steps over long time scales. In Brownian dynamics models, these constraints lead to linear equations which represent saddle point problems describing the velocity tangent to the constraint manifold and the forces resulting from imposition of the constraint on the dynamics. Simulations of these constrained systems must solve these saddle point problems, which are often ill-conditioned and require clever preconditioning in order to facilitate large scale simulation using iterative methods like GMRES. In this work, we show that the projected conjugate gradient (PrCG) method, which applies conjugate gradients projected onto the tangent space of the constraint manifold, has certain advantages compared to preconditioned iterative methods like GMRES, without the need to devise and compute a preconditioner. Specifically, for simulations of beads interacting hydrodynamically with commonly encountered holonomic constraints, PrCG exhibits linear scaling with the number of beads similar to GMRES. However, because PrCG does not require a preconditioner it often requires less precomputational effort both in terms of computational complexity and storage, depending on the nature of the constraints. Furthermore, PrCG produces iterates which are always feasible, allowing for independent tuning of the error tolerance on the bead velocities from the error tolerance on the constraint equations. We demonstrate PrCG's efficiency by solving several example low Reynolds number systems involving rigid bodies, freely jointed chains, and immobile particles, as well as a mixed constraint problem with both a rigid body and immobile particles, comparing to GMRES and analytical solutions, when available.