n this thesis, we study preconditioned iterative methods for the linear systems. This system of linear equations arising in the numerical integration of ODEs and time-dependent PDEs by implicit Runge-Kutta and boundary value methods. A preconditioning strategy based on a Kronecker product splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matrix are established. The proposed KPS method is proven to be convergent under some conditions and the optimal parameter is dependent on the integration schemes but is independent of the ODEs and the stepsizes. In the end, Numerical examples are presented to illustrate the effectiveness of this approach.