School of Computer Science and Applied Mathematics (ETDs)

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    Overlapping multidomain paired quasilinearization methods for solving boundary layer flow problems
    (University of the Witwatersrand, Johannesburg, 2024) Nefale, Mpho Mendy; Otegbeye, Olumuyiwa; Oloniiju, Shina Daniel
    There is a constant and continuous need to refine current numerical approaches used to solve non-linear differential equations, which are employed to model real- world problems that often do not have analytical solutions. Spectral-based techniques have proven to be one of the most efficient numerical techniques for finding solutions of differential equations. Numerous spectral-based linearization techniques have been developed, such as the spectral relaxation (SRM), the spectral local linearization (SLLM), the spectral quasilinearization (SQLM), and the paired quasilinearization (PQLM) methods, among others. Previous research suggests that the PQLM is an efficient approach for solving complex non-linear systems of ordinary (ODEs) and partial differential equations (PDEs). However, it has been observed that this method requires further enhancement when utilized for problems described over a large domain, be it temporal or spatial. This research aims to address this limitation by proposing a modified version of the PQLM called the overlapping multi-domain paired quasilinearization method (OMD-PQLM), that enhances the accuracy and convergence speed of the original approach. The new approach entails solving a system by a technique that involves decoupling the system into pairs of equations and partitioning the large domain into smaller overlapping sub-domains. A comparison between the OMD-PQLM and the PQLM is conducted by solving systems of ODEs and PDEs. The proposed numerical approach is evaluated based on the norms of the residual and convergence errors, computational time, and the influence of the number of grid points and sub-domains on the convergence speed of the iterative scheme and the accuracy of the solutions. The findings demonstrate that the OMD-PQLM remarkably improves the accuracy of the solution compared to the PQLM, suggesting that partitioning the problem domain into overlapping multiple-domains optimizes the performance of the PQLM.