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Abstract
This thesis is divided into two parts: The first introduces a new time integration framework that is based on interpolating polynomials, and the second extends exponential integration to the spectral deferred correction framework. Both parts discuss time integration methods that can be derived without solving nonlinear order conditions.
In part I, we introduce a time-integration framework for solving systems of first-order ordinary differential equations by using interpolating polynomials. Our approach is to combine ideas from complex analysis and approximation theory to construct new integrators. This strategy allows us to trivially satisfy order conditions and easily construct a range of implicit or explicit integrators with properties such as parallelism and high-order of accuracy. The breadth of our framework is made possible by combining ideas from complex analysis, approximation theory, and general linear methods. In this work, we present several example polynomial methods including generalizations of the backward differentiation formula and Adams-Moulton methods. We compare the stability regions of these generalized methods to their classical counterparts and find that the new methods offer improved stability especially at high order. Finally, we evaluate the performance of our polynomial integrators by running a variety of numerical experiments and find that polynomial integrators offer improved stability and accuracy in comparison to classical integrators.
In part II, we introduce a new class of arbitrary-order exponential time differencing methods based on spectral deferred correction and describe a simple procedure for initializing the requisite matrix functions. We compare the stability and accuracy properties of our new exponential methods to those of an existing implicit-explicit spectral deferred correction scheme. We find that exponential integrators have larger accuracy regions and comparable stability regions. We conduct numerical experiments to compare exponential and implicit-explicit spectral deferred correction schemes against a competing fourth-order exponential Runge-Kutta scheme. We find that high-order exponential spectral deferred correction schemes are the most efficient in terms of function evaluations and overall speed when solving partial differential equations to high accuracy. Our results suggest that high-order exponential spectral deferred correction schemes are well-suited to work in conjunction with spectral spatial methods or other high-order spatial discretizations.
