Exact Discrete Solutions of Boundary Control Problems for the 1D Heat Equation

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.