论文标题
使用精确离散溶液的1D热方程的硬摩尔边界控制问题
A Stiff MOL Boundary Control Problem for the 1D Heat Equation with Exact Discrete Solution
论文作者
论文摘要
法线离散化是对硬集成方法的要求解决问题。但是，对于已知分析解决方案的PDE问题，空间离散误差的存在或使用代码来计算参考解决方案可能会限制数值测试结果的有效性。为了克服这些缺点，我们在此简短说明了一个简单的测试问题，即边界控制，这种情况可能会降低订单。我们得出了由一维离散热方程和目标函数控制的最佳边界控制问题解决方案的精确公式，该公式可以测量最终状态与目标和控制成本的距离。这种分析设置用于比较选定的隐式runge-kutta和对等经典阶订单四步方法的数值观察到的收敛顺序，这些方法适用于最佳控制问题。
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.
