Runge-Kutta-Nystrom Methods For Solving Oscillatory Problems

New Runge-Kutta-Nyström (RKN) methods are derived for solving system of second-order Ordinary Differential Equations (ODEs) in which the solutions are in the oscillatory form. The dispersion and dissipation relations are imposed to get methods with the highest possible order of dispersion and dissipation. The derivation of Embedded Explicit RKN (ERKN) methods for variable step size codes are also given. The strategies in choosing the free parameters are also discussed. We analyze the numerical behavior of the RKN and ERKN methods both theoretically and experimentally and comparisons are made over the existing methods. In the second part of this thesis, a Block Embedded Explicit RKN (BERKN) method are developed. The implementation of BERKN method is discussed. The numerical results are compared with non block method. We find that the new code on Block Embedded Explicit RKN (BERKN) method is more efficient for solving system of second-order ODEs directly. Next, we discussed the derivation of Diagonally Implicit RKN (DIRKN) methods for solving stiff second order ODEs in which the solutions are oscillating functions. The dispersion and dissipation relations are developed and again are imposed in the derivation of the methods. For solving oscillatory problems with high frequency, method with P-stability property is discussed. We also derive the Embedded Diagonally Implicit RKN (EDIRKN) methods for variable step size codes. To see the preciseness and effectiveness of the methods, the constant and variable step size codes are developed and numerical results are compared with current methods given in the literature. Finally, the Parallel Embedded Explicit RKN (PERKN) method is developed. The parallel implementation of PERKN on the parallel machine is discussed. The performance of the PERKN algorithm for solving large system of ODEs are presented. We observe that the PERKN gives the better performance when solving large system of ODEs. In conclusion, the new codes developed in this thesis are suitable for solving system of second-order ODEs in which the solutions are in the oscillatory form.

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