Implementation of second derivative general linear methods

In this paper, the implementation of second derivative general linear methods (SGLMs) in a variable stepsize environment using Nordsieck technique is discussed and various implementation issues are investigated. All coefficients of a method of order four together with its error estimate are obtained. The method is derived with the aim of good zero-stability properties for a large range of ratios of sequential stepsizes to implement in a variable stepsize mode. The numerical experiments indicate that the constructed error estimate is very reliable in a variable stepsize environment and beautifully confirm the efficiency and robustness of the proposed scheme based on SGLMs. Moreover, the results verify that the proposed scheme outperforms the code ode15s from Matlab ODE suite on systems whose Jacobian has eigenvalues which are close to the imaginary axis.