Peer Two-Step Methods with Embedded Sensitivity Approximation for Parameter-Dependent ODEs

Abstract

Peer two-step methods have been successfully applied to initial value problems for stiff and nonstiff ordinary differential equations (ODEs) both on parallel and sequential computers. Their essential property is the use of several stages per time step with the same accuracy. As a new application area these methods are now used for parameter-dependent ODEs where the peer stages approximate the solution also at different places in the parameter space. The main interest here is sensitivity data through an approximation of solution derivatives in different parameter directions. Basic stability and convergence properties are discussed and peer methods of order 2 and 3 in the time stepsize are constructed. The computed sensitivity matrix is used in approximate Newton and Gauss--Newton methods for shooting in boundary value problems, where initial values and/or ODE parameters are searched for, and in parameter identification from partial information on trajectories.