A splitting preconditioner for implicit Runge-Kutta discretizations of a partial differential-algebraic equation

In this paper we study efficient iterative methods for solving the system of linear equations arising from the fully implicit Runge-Kutta discretizations of a class of partial differential-algebraic equations. In each step of the time integration, a block two-by-two linear system is obtained and needed to be solved numerically. A preconditioning strategy based on an alternating Kronecker product splitting of the coefficient matrix is proposed to solve such linear systems. Some spectral properties of the preconditioned matrix are established and numerical examples are presented to demonstrate the effectiveness of this approach.