Explicit hybrid methods for solving special second order ordinary differential equations

The focus of this thesis is to derive new two-step explicit hybrid methods for the numerical solution of system of special second order ordinary differential equations of the form y"=f ( x, y) . Explicit hybrid methods of order seven have been developed by employing strategies of selecting free parameters. Dissipation relations are imposed to obtain a method with highest possible order of dissipation. Phase-lag and stability analysis are presented. Numerical results show that the methods give better accuracy compared with the existing methods. For variable step-size codes,embedded pairs of explicit hybrid methods are introduced. The phase-lag and stability interval of the methods are given and the procedure of controlling the stepsize change is described. To improve the accuracy of hybrid methods, the construction of exponentially fitted explicit hybrid methods is investigated. The derivations of the methods with two stages and four stages are described in detail. The method with two stages is derived for constant step-size code whereas the method with four stages is derived for variable step-size code. Their stability regions and the numerical results are given. Finally, the construction of a block explicit hybrid method implemented on a parallel computer is discussed. This method calculates two consecutive points using two independent formulas. The stability analysis of the formula which computes the second point is presented. The parallel implementation of the method is evaluated in terms of accuracy and speedup. From the results, it is observed that the speedup is greater than 1.5 which indicates that the parallel code is faster than the sequential one. On the whole, this study reveals that the new methods are capable and efficient for solving special second order ordinary differential equations.

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