The possibility of constructing a conservative numerical method for the solution of the cauchy problem for Hamiltonian systems based on two-stage symmetric symplectic Runge–Kutta methods

We study the possibility of constructing a computational method for solving the Cauchy problem for Hamiltonian systems, which yields an approximate solution satisfying the law of total energy conservation. The presented technique is based on a family of two-stage symmetric symplectic Runge–Kutta methods. The properties of the proposed method are investigated for the case of a model problem on a material point motion in a cubic field potential. The possibility of working out the method yielding a numerical solution, which preserves the total energy over the period of the problem finite solution except for small neighborhoods of cusps. Time dependences of the symplecticity and reversibility defects are studied on the numerical solution obtained by the constructed method.