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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.

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Metadaten
Author:Jens Lang, Bernhard A. Schmitt
DOI:https://doi.org/https://doi.org/10.1007/s10957-022-02154-4
Document Type:Article
Language:English
Date of Publication (online):2024/03/21
Release Date:2024/03/25
Volume:Journal of Optimization Theory and Applications
Issue:Vol. 196
Page Number:13
First Page:1106
Last Page:1118
Institutes:Technische Universität Darmstadt
Subprojects:B01
Licence (German):License LogoCreative Commons - CC BY - Namensnennung 4.0 International