Many-body dynamics problems are expected to handle millions of unknowns when, for instance, investigating the three-dimensional flow of granular material. Unfortunately, the size of the problems tractable by existing numerical solution techniques is severely limited on convergence grounds. This is typically the case when the equations of motion embed a differential variational inequality problem that captures contact and possibly frictional interactions between rigid and/or flexible bodies. As the size of the physical system increases, the speed and/or the quality of the numerical solution decreases. This paper describes three methods – the gradient projected minimum residual method, the preconditioned spectral projected gradient with fallback method, and the modified proportioning with reduced gradient projection method – that demonstrate better scalability than the projected Jacobi and Gauss–Seidel methods commonly used to solve contact problems that draw on a differential-variational-inequality-based modeling approach. Copyright © 2013 John Wiley & Sons, Ltd.