Two-Step Symmetrization of Initial Value Problems

Abstract

In this thesis we study two-step symmetrization of Runge–Kutta methods for stiff ordinary differential equations. The process is a generalization of the smoothing formula introduced by Gragg in 1965. It can be regarded as being generated by a related Runge–Kutta method constructed so as to preserve the asymptotic error expansion in even powers of the stepsize. When the method is applied with an accelerating technique such as extrapolation, the order can then be increased by two at a time. The method is also L-stable and hence provides damping for stiff problems. A study of one-step symmetrization by Gorgey and Chan in 2012 has shown how one-step symmetrization can be applied in active and passive modes and combined with active or passive extrapolation. Two-step symmetrization has twice the number of parameters as one-step symmetrization, thus allowing greater flexibility in satisfying order and other desirable conditions. This provides motivation for us to investigate two-step symmetrization and what advantages this extension has over one-step symmetrization. With two-step symmetrization, the implicit midpoint rule and the implicit trapezoidal rule retain their order-2 behaviour whereas with one-step symmetrization these methods behave like an order-1 method in the active mode. We also observe that the two-step symmetrization of the implicit trapezoidal rule shows superconvergent order-4 behaviour for both active and passive symmetrizations when solving certain stiff linear problems. In this study we apply the symmetrization process in both constant and variable stepsize settings and test the efficiency and accuracy of the methods for linear and nonlinear problems. We also extend the application to the higher order 2-stage Gauss method of order 4 and show how the symmetrizer is able to suppress order reduction and thus obtain order-6 behaviour with extrapolation.