Order-4 symmetrized runge-kutta methods for stiff problems (Kaedah Runge-Kutta Tersimetri Peringkat-4 untuk Masalah Kaku)

Abstract

If a Runge-Kutta method having an asymptotic error expansion in the stepsize h is symmetric then it is characterised by an h2-expansion. Since elimination of the leading error terms in succession results in an increase in the order by two at a time, a symmetric method could therefore be suitable for the construction of extrapolation methods. However, when order reduction occurs for stiff problems it needs to be suppressed before an appropriate extrapolation formula can be applied. This can be achieved by a process called symmetrization which is a composition of the symmetric method with an L-stable method known as a symmetrizer. The symmetrizer is constructed so as to preserve the h2-asymptotic error expansion. In this paper we consider symmetrization of the 2-stage Gauss and the 3-stage Lobatto IIIA methods of order 4. We show that these methods are more efficient when used with symmetrization. Extrapolation based on the symmetrized methods is therefore expected to give greater accuracy. We also show that the method with a higher stage order is more advantageous than one with a lower stage order for solving stiff problems.