Peer methods with improved embedded sensitivities for parameter-dependent ODEs

Recently, an extension of peer two-step methods to parameter-dependent IVPs has been proposed by the author and E. Kostina where additional embedded stages allow cheap computation of solution derivatives (sensitivities) with respect to problem parameters. For each parameter only one additional ‘satellite’ stage was used which provides a high-order approximation to a neighboring solution. However, the accuracy of the sensitivities was only first order in the time stepsize hh. In the present paper we derive implicit peer methods where the sensitivity accuracy is of second order with the same number of stages. Approximate implementations yield efficient linearly-implicit methods for stiff problems and explicit predictor–corrector-type versions for nonstiff problems. Tests confirm improved and more robust convergence of inexact Newton iterations in shooting for different applications with nonstiff and stiff IVPs.