Exact Discrete Solutions of Boundary Control Problems for the 1D Heat Equation
针对一维离散热方程的最优边界控制问题，推导了精确解公式，用于测试隐式龙格-库塔和Peer两步法在阶数降低情况下的收敛阶。
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.