A Stiff MOL Boundary Control Problem for the 1D Heat Equation with Exact Discrete Solution
- Method-of-lines discretizations are demanding test problems for stiff inte-
gration 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 over-
come 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 numeri-
cally observed convergence orders for selected implicit Runge-Kutta and Peer
two-step methods of classical order four which are suitable for optimal control
problems.