A Stiff MOL Boundary Control Problem for the 1D Heat Equation with Exact Discrete Solution

TL;DR

This paper introduces an exact discrete solution for a 1D heat equation boundary control problem, providing a benchmark to evaluate the convergence of numerical methods without discretization errors.

Contribution

It derives exact formulas for the solution of a boundary control problem, enabling precise assessment of numerical methods' convergence orders.

Findings

Exact formulas for the solution are derived.

Numerical methods' convergence orders are accurately assessed.

One-step methods may experience order reduction in this setting.

Abstract

Method-of-lines discretizations are demanding test problems for stiff integration methods. However, for PDE problems with known analytic solution the presence of space discretization errors or the need to use codes to compute reference solutions may limit the validity of numerical test results. To overcome these drawbacks we present in this short note a simple test problem with boundary control, a situation where one-step methods may suffer from order reduction. We derive exact formulas for the solution of an optimal boundary control problem governed by a one dimensional discrete heat equation and an objective function that measures the distance of the final state from the target and the control costs. This analytical setting is used to compare the numerically observed convergence orders for selected implicit Runge-Kutta and Peer two-step methods of classical order four which are suitable for optimal control problems.