Topics in the numerical analysis of ordinary differential equations: Molecular dynamics and chemical kinetics.

摘要

This thesis consists of four independent chapters. The first three chapters develop and investigate some simple test problems for molecular dynamics simulations. Each of the chapters examines a deterministic system that approximates a stochastic process in some limit. The motivation is to provide systems which can be used to evaluate the ability of numerical methods to correctly compute statistical properties of systems. The last chapter presents an algorithm for the integration of the equations of chemical kinetics.; The first chapter considers two Hamiltonian systems of small numbers of particles interacting on compact domains. The ability of different numerical integrators to reproduce the statistical features of the system is investigated. It is observed that symplectic methods reproduce statistics quite well, whereas step-and-project methods produce erroneous statistical information. An analysis based on Markov chain models is presented.; The second chapter considers a deterministic system of many particles interacting on a finite interval. It is shown that a tracer particle in this system has a trajectory that converges to a stationary Gaussian process. Numerical experiments are performed on a modification of this system using the Symplectic Euler method. The method is observed to reproduce the same stochastic limit.; In the third chapter, it is shown how stationary Gaussian processes, particularly the Ornstein-Uhlenbeck process, can be approximated by deterministic systems with random data. Such processes are obtained as the limit of a sequence of uncoupled oscillator systems. Weak convergence is proven on C[0, T] and the convergence of a wide variety of long-term averages is also shown to hold.; In the last chapter, an adaptive algorithm for the model reduction of chemical systems is presented. The algorithm works in tandem with a numerical integrator to reduce the complexity of the system as the kinetic equations are being solved. It allows some systems to be integrated more quickly than with straightforward integration.