Active and passive symmetrization of Runge–Kutta Gauss methods

A symmetrizer for a symmetric Runge–Kutta method is designed to preserve the asymptotic error expansion in even powers of the stepsize and to provide damping for stiff initial value problems. In this paper we study symmetrizers for the Gauss methods with two and three stages and compare the implementation in passive and active modes. In particular, we perform a detailed analysis of the Prothero–Robinson problem which provides insight into the behaviour of symmetrizers in suppressing order reduction experienced by the symmetric methods. We present numerical results on the effects of passive and active symmetrization for some stiff linear and nonlinear problems. These effects have important implications for the development of extrapolation methods based on higher order symmetric methods for the numerical solution of stiff problems. Our results show that symmetrization in both modes improves accuracy and efficiency, and can restore the classical order of the Gauss methods for stiff linear problems. We compare the two modes of symmetrization for constant stepsize and present preliminary results in a variable stepsize setting.