Extrapolation of Runge-Kutta methods for stiff initial value problems

Abstract

Extrapolation methods provide one of the important types of numerical integrators for ordinary differential equations with an efficient stepsize control mechanism and a simple variable order strategy. Those based on symmetric discretizations which possess an asymptotic error expansion in even powers of the stepsize are therefore particularly attractive. Runge-Kutta methods have advantages in parallel computing and are self-starting. Implicit ones have strong stability properties and are therefore good candidates for stiff problems. Combining these advantages, extrapolation methods based on implicit Runge-Kutta formulae are thus suitable for the numerical solution of stiff initial value problems. In this thesis we study the properties of these methods. The A-stability of extrapolations based on Runge-Kutta methods of high order are investigated and several barrier results obtained. An algebraic characterization of symmetry equivalent to that first given by Stetter is presented and used to derive a one-parameter family of algebraically stable symmetric methods based on Lobatto quadrature of order 2s - 2. Extrapolations of arbitrary symmetric methods are shown not to be A.-stable. The characterization of symmetry is generalized. Several families of composite methods which are not symmetric according to the characterization of Stetter but which preserve asymptotic error expansions in even powers of the stepsize are constructed. Certain extrapolations of these generalized symmetric methods are shown to be ,A-stable. The properties (order, linear and nonlinear stability, smoothing and damping, order defect phenomenon, convergence and asymptotic error expansions) of these generalized symmetric methods are studied. Special attention is focussed on the methods based on the implicit midpoint and implicit trapezoidal rules as well as the 2-stage Gauss method of order 4 for certain stiff model problems, particularly in the strongly stiff case.