Variable-Stepsize Implicit Peer Triplets in ODE Constrained Optimal Control

TL;DR

This paper develops and analyzes super-convergent implicit Peer two-step methods with variable stepsizes for ODE-constrained optimal control, ensuring high order accuracy and stability across varying time scales.

Contribution

It introduces upgraded variable-stepsize implicit Peer triplets with new order conditions, and proves their convergence and practical effectiveness in optimal control problems.

Findings

Achieved super-convergence for variable stepsize Peer methods.

Constructed 4-stage Peer triplets with specific order pairs.

Numerical tests confirm the theoretical order of convergence.

Abstract

This paper is concerned with the theory, construction and application of implicit Peer two-step methods that are super-convergent for variable stepsizes, i.e., preserve their classical order achieved for uniform stepsizes when applied to ODE constrained optimal control problems in a first-discretize-then-optimize setting. We upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to get ready for dynamical systems with varying time scales without loosing efficiency. Peer triplets consist of a standard Peer method for interior time steps supplemented by matching methods for the starting and end steps. A decisive advantage of Peer methods is their absence of order reduction since they use stages of the same high stage order. The consistency analysis of variable-stepsize implicit Peer methods results in additional order conditions and severe new difficulties for uniform zero-stability, which intensifies the demands on the Peer triplet. Further, we discuss the construction of 4-stage methods with order pairs (4,3) and (3,3) for state and adjoint variables in detail and provide four Peer triplets of practical interest. We rigorously prove convergence of order $s-1$ for $s$-stage Peer methods applied on grids with bounded or smoothly changing stepsize ratios. Numerical tests show the expected order of convergence for the new variable-stepsize Peer triplets.