Sequential and parallel Modified Accelerated Overrelaxation iterative methods for solving integer and fractional order partial differential equations with multi-core architecture

This thesis deals with sequential and parallel Modified Accelerated Overrelaxation (MAOR) iterative methods for solving the Helmholtz equation and Fractional Poisson equation. The thesis examined different finite-difference approximations, in the form of point and group approaches, that are discretized on the second order, fourth order and fractional order iterative methods. The utilization of relaxation schemes along with the finite-difference approximation would increase the convergent rate of the it- erative methods. This thesis would consider the MAOR scheme which was developed as an extension of the Accelerated Overrelaxation (AOR) scheme and is compared to some previously developed relaxation schemes. The OpenMP parallel program was implemented together with the iterative methods and relaxation schemes which would reduce the execution time of the iterative methods while maintaining the number of iterations. To assist this process, different ordering strategies are applied to the solu- tion domain which ensured that the points in the iterative process are independent from each other. Numerical experiments showed that the MAOR scheme produced the least number of iteration when compared to other relaxation schemes. The implementation of parallel computing on all iterative methods has reduced the execution time when the number of processors increases which also increased the parallel efficiency. However, the MAOR scheme utilizes more points during the iterative process which contributed to having a higher execution time compared to some relaxation schemes.

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