Extrapolation of Symmetrized Runge-Kutta Methods

Abstract

Smoothing was first introduced by Gragg in 1964 for ordinary differential equations and later used by others in one-step methods of order 1 or 2 for stiff equations. The generalization of smoothing to symmetric Runge-Kutta methods of arbitrary order is called symmetrization and was due to Chan. It involves an L-stable method, called a symmetrizer, constructed to preserve the h2-asymptotic error expansion of an associated symmetric method. The resulting damping suppresses order reduction in solving stiff problems and thus increases the efficiency of extrapolation by improving the accuracy by two orders at a time. In this thesis we study symmetric methods of orders 4 and 6 from the Gauss and Lobatto IIIA families and investigate the accuracy and efficiency when applied with symmetrization and extrapolation. The emphasis of this thesis is on implementation of the numerical methods and numerical experiments. In the constant stepsize setting, we study two ways of applying symmetrization, active and passive, and four ways of applying extrapolation with symmetrization. We give a complete theoretical analysis of the order behaviour for the Prothero-Robinson problem and verify the results by numerical experiments. We observe that the symmetrized Gauss methods are more efficient than the symmetrized Lobatto IIIA methods of the same order for stiff linear problems. However, for two dimensional nonlinear problems, the symmetrized 4-stage Lobatto IIIA method is more efficient. In all cases, we observe numerically that passive symmetrization with passive extrapolation is more efficient than active symmetrization with active extrapolation. In the variable stepsize setting, we find that symmetrization can be used for error estimation. We find that this error estimation is more efficient than the use of the local extrapolation approach. There are two ways of applying extrapolation with symmetrization in the variable stepsize setting. In the numerical experiments on the STIFF DETEST problem set we observe that passive symmetrization with active extrapolation is more efficient than active symmetrization with active extrapolation. For linear problems the symmetrized 3-stage Gauss method with active extrapolation is more efficient that the symmetrized 4-stage Lobatto IIIA method with active extrapolation. For nonlinear problems, however, the opposite is observed. This thesis provides definite evidence that symmetric methods of high order when applied with appropriate symmetrization/extrapolation can be effective practical methods for the numerical solution of stiff linear and nonlinear problems.