Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order (Q5934384)

For a second order stiff initial value problem having periodic solutions, the authors investigate the construction of four-stage fourth-order symplectic and P-stable singly diagonally implicit Runge-Kutta-Nyström (SDIRKN) methods which have high order of dispersion (order 8). Conditions of symplecticness, order conditions for order 4, conditions of symmetry and conditions of stability are imposed and are solved to obtain such methods. Numerical experiments are considered for four test problems and numerical results show that SDIRKN methods perform very efficiently.