On the possibility of constructing a conservative numerical method of solution of the initial value problem for Hamiltonian systems on the basis of the two-stage symmetric symplectic Runge–Kutta methods

The question about the possibility of working out a Hamiltonian system initial value problem solution numerical method which gives an approximate solution that conforms the total energy conservation law is studied in the work. The method is constructed on basis of the family of two-stage symmetric symplectic Runge–Kutta methods. The properties of the suggested method are studied by the example of the model problem about the particle motion in the cubic potential field. The possibility of working out the method giving a numerical solution which preserves the total energy on the period of the problem finite solution, except for small neighbourhood of return points, is shown. The time dependencies of the symplecticity and reversibility defects on the numerical solution obtained by the worked out method are investigated.