Explicit Runge-Kutta-Nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation

An explicit Runge-Kutta-Nyström (RKN) method with high order dispersion (phase-lag) and dissipation (amplification error) properties is studied for the integration of initial-value problems (IVP) of second-order ordinary differential equations (ODEs) possessing oscillating solutions. The constructions of RKN methods for constant step size and embedded RKN pair for variable step size have been derived. The effects of dispersion and dissipation relations are tested on homogeneous and non-homogenous test problems which have oscillatory solutions.Derivation of symplectic explicit Runge-Kutta-Nyström method is studied for Hamiltonian system with oscillating solutions. Symplectic methods can be more efficient than non-symplectic methods for long interval of integration. Numerical results show that the symplectic methods with high order of dispersion are more efficient for solving second order ordinary differential equations. The new fourth and fifth order explicit RKN methods with dispersion (phase-fitted) and dissipation (amplification-fitted) of order infinity have been derived. The fifth order explicit RKN methods is divided into two parts; methods with phase-fitted and methods with both phase-fitted and amplification-fitted. In this thesis, the phase-fitted methods are derived based on symplectic method by Sharp and method derived in this thesis. For fifth order phase-fitted and amplification-fitted consists of four- and five-stage RKN methods with First Same As Last (FSAL) technique. Numerical results show that our methods are much more efficient than existing method with the same algebraic order. In conclusion, we have derived explicit RKN methods with dispersion and dissipation for solving second-order ODEs that possessing oscillatory solutions. The dispersion constant, dissipation constant and local truncation error (LTE) terms are also calculated. The homogeneous and non-homogenous test problems with oscillatory solutions are used to prove the efficiency of our methods.

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