A leapfrog multigrid algorithm for the optimal control of parabolic PDEs with Robin boundary conditions

We develop a second-order finite difference scheme for solving the first-order necessary optimality systems arising from the optimal control of parabolic PDEs with Robin boundary conditions. Under the framework of matrix analysis, the proposed leapfrog scheme is shown to be unconditionally stable and second-order convergent for both time and spatial variables, without the requirement of the classical Courant–Friedrichs–Lewy (CFL) condition on the spatial and temporal mesh step sizes. Moreover, the developed leapfrog scheme provides a well-structured discrete algebraic system that allows us to establish an effective multigrid iterative fast solver. The resultant multigrid solver demonstrates a mesh-independent convergence rate and a linear time complexity. Numerical experiments are provided to illustrate the accuracy and efficiency of the proposed leapfrog scheme.