Multi-scale methods in time and space for particle simulations

Particle simulations arise in many applications such as gravitational N -body problems, electrostatics, and molecular dynamics simulations. The motion of the particles is governed by Newton’s equations of motion. Since the system of equations is typically non-linear, its solution requires numerical simulation via time integration. During each integration step the force on each particle needs to be computed. If long-range forces such as gravity or electrostatics are present the direct evaluation of the force for an N -particle system is an O(N) procedure which quickly becomes intractable for large systems. Two common approaches for improving the computational efficiency of the force evaluation, and hence particle simulations, are computing the force less frequently and computing the force more efficiently. In this dissertation we discuss how multi-scale methods in time and space can be applied towards this end. One way to compute the force less often is to assign a larger time step to the longrange forces. While the idea of multiple-time-stepping (MTS) has been explored in previous works, we will formulate a class of integrators called asynchronous variational integrators (AVI) in the context of particle simulations that allows for time steps to be chosen in arbitrary ratio. Although MTS schemes can be used to reduce the cost of particle simulations, there are potential drawbacks such as the presence of instabilities whose manifestation is dependent on the choice of time steps. A thorough linear stability analysis of AVI is performed to explore this issue. AVI is also extended to accommodate Langevin dynamics which is commonly used as a stochastic thermostat for molecular dynamics simulations of proteins. While using multiple time steps reduces the cost of force evaluations, this reduction