An efficient implicit Runge–Kutta method for second order systems

We will consider the efficient implementation of a fourth order two stage implicit Runge–Kutta method to solve periodic second order initial value problems. To solve the resulting systems, we will use the factorization of the discretized operator. Such proposed factorization involves both complex and real arithmetic. The latter case is considered here. The resulting system will be efficient and small in size. It is one fourth the size of systems using normal implicit Runge–Kutta method. Numerical details and examples will also be presented to demonstrate the efficiency of the method.