Extrapolation of symmetrized Runge-Kutta methods in the variable stepsize setting

摘要

Symmetrization technique is shown to be efficient when applied with extrapolation for the Gauss and Lobatto IIIA methods in the constant stepsize setting. The most efficient way of applying symmetrization in this constant stepsize setting is the passive symmetrization with passive extrapolation. On the other hand, in the variable stepsize setting, passive symmetrization combined with active extrapolation is shown to be more efficient than with active symmetrization. This article focuses on the variable stepsize setting for the 3-stage Gauss and 4-stage Lobatto IIIA methods where the error estimation plays an important role. The error estimation for the 3-stage Gauss and 4-stage Lobatto IIIA methods are estimated using the symmetrization approach. This approach is compared with the stepsize doubling technique and the numerical results are tested for the Van der Pol, Robertson and Oregonator problems. It is observed that the symmetrization approach to estimate the error gives the most efficient results. In addition to this, two solvers with the symmetrized 3-stage Gauss and the symmetrized 4-stage Lobatto IIIA methods with extrapolation known as SYMEXa and SYMEXb respectively are tested for the stiff DETEST problems. These solvers use the symmetrization approach to estimate the local error. The efficiencies of these solvers are compared with the RADAU5 solver with extrapolation that uses the step-size doubling approach. The numerical results shows that for certain classes of problems, SYMEXa and SYMEXb are more efficient than the RADAU5 method.