A Lyapunov exponent based stability theory for ordinary differential equation initial value problem solvers

Abstract

In this dissertation we consider the stability of numerical methods approximating the solution of bounded, stable, and time-dependent solutions of ordinary differential equation initial value problems. We use Lyapunov exponent theory to determine conditions on the maximum allowable step-size that guarantees that a one-step method produces a decaying numerical solution to an asymptotically contracting, time-dependent, linear problem. This result is used to justify using a one-dimensional asymptotically contracting real-valued nonautonomous linear test problem to characterize the stability of a one-step method. The linear stability result is applied to prove a stability result for the numerical solution of a class of stable nonlinear problems. We use invariant manifold theory to show that we can obtain similar stability results for strictly stable linear multistep methods approximating asymptotically contracting, time-dependent, linear problems by relating their stability to the stability of an underlying one-step method. The stability theory for one-step methods is used to devise a procedure for stabilizing a solver that fails to produce a decaying solution to a linear problem when selecting step-size using standard error control techniques. Additionally, we develop an algorithm that selects step-size for the numerical solution of a decaying nonautonomous scalar test problem based on accuracy and the stability theory we developed.