loading page

Crank-Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problem
  • Caijie Yang,
  • Tongjun Sun
Caijie Yang
Shandong University

Corresponding Author:[email protected]

Author Profile
Tongjun Sun
Shandong University
Author Profile


In this paper, we adopt the optimize-then-discretize approach to solve parabolic optimal Dirichlet boundary control problem. First, we derive the first-order necessary optimality system, which includes the state, co-state equations and the optimality condition. Then, we propose Crank-Nicolson finite difference schemes to discretize the optimality system in 1D and 2D cases, respectively. In order to build the second order spatial approximation, we use the ghost points on the boundary in the schemes. We prove that the proposed schemes are unconditionally stable, compatible and second-order convergent in both time and space. To avoid solving the large coupled schemes directly, we use the iterative method. Finally, we present a numerical example to validate our theoretical analysis.
03 Aug 2021Submitted to Mathematical Methods in the Applied Sciences
04 Aug 2021Submission Checks Completed
04 Aug 2021Assigned to Editor
16 Aug 2021Reviewer(s) Assigned
04 Nov 2021Review(s) Completed, Editorial Evaluation Pending
14 Nov 2021Editorial Decision: Revise Minor
04 Dec 20211st Revision Received
06 Dec 2021Submission Checks Completed
06 Dec 2021Assigned to Editor
06 Dec 2021Reviewer(s) Assigned
04 Mar 2022Review(s) Completed, Editorial Evaluation Pending
05 Mar 2022Editorial Decision: Accept
16 Mar 2022Published in Mathematical Methods in the Applied Sciences. 10.1002/mma.8244