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

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

Backward Error Analysis as a Model of Computation for Numerical Methods

This thesis delineates a generally applicable perspective on numerical meth­ ods for scientific computation called residual-based a posteriori backward er­ ror analysis, based on the concepts of condition, backward error, and residual, pioneered by Turing and Wilkinson. The basic underpinning of this perspec­ tive, that a numerical method’s errors should be analyzable in the same terms as physical and modelling errors, is readily understandable across scientific fields, and it thereby provides a view of mathematical tractability readily in­ terpretable in the broader context of mathematical modelling. It is applied in this thesis mainly to numerical solution of differential equations. We examine the condition of initial-value problems for ODEs and present a residual-based error control strategy for methods such as Euler’s method, Taylor series meth­ ods, and Runge-Kutta methods. We also briefly discuss solutions of continuous chaotic problems and stiff problems

Noise induced changes to dynamic behaviour of stochastic delay differential equations

This thesis is concerned with changes in the behaviour of solutions to parameter-dependent stochastic delay differential equations

Pseudospectral reduction to compute Lyapunov exponents of delay differential equations

A recent pseudospectral collocation is used to reduce a nonlinear delay differential equation to a system of ordinary differential equations. Standard methods are then applied to compute Lyapunov exponents. The validity of this simple approach is shown experimentally. Matlab codes are also included

Time elements for enhanced performance of the Dromo orbit propagator

We propose two time elements for the orbit propagator named Dromo. One is linear and the other constant with respect to the independent variable, which coincides with the osculating true anomaly in the Keplerian motion. They are defined from a generalized Kepler’s equation written for negative values of the total energy and, unlike the few existing time elements of this kind, are free of singularities. To our knowledge it is the first time that a constant time element is associated with a second-order Sundman time transformation. Numerical tests to assess the performance of the Dromo method equipped with a time element show the remarkable improvement in accuracy for the perturbed bounded motion around the Earth compared to the case in which the physical time is a state variable. Moreover, the method is competitive with and even better than other efficient sets of elements. Finally, we also derive a time element for a null and positive total energy