Leapfrog multigrid methods for parabolic optimal control problems

A second-order leapfrog finite difference scheme in time is proposed to solve the first-order necessary optimality systems arising from parabolic optimal control problems. Different from classical approximation, the proposed leapfrog scheme appears to be unconditionally stable. More importantly, the developed leapfrog scheme provides a well-structured discrete algebraic system and allows us to establish a fast linear solver under the multigrid framework. The unconditional stability of the scheme is proved under the L² norm. Numerical results show that our presented scheme significantly outperforms the widely used Crank-Nicolson scheme and the resultant fast solver demonstrates a mesh-independent convergence rate as well as a desirable feature of linear time complexity.