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  • The Quotient Multi-Grid Reduction in Time (QMGRIT) algorithm is developed and applied in this dissertation, showing its potential to solve intricate scientific and engineering problems like electric motor modelling. We investigate the efficacy of the QMGRIT algorithm in solving periodic time-dependent partial differential equations (PDEs), with particular attention to the heat and wave equations, aiming to scope both kinds of PDEs: the parabolic and hyperbolic PDEs. We validate the QMGRIT algorithm by constructing its mathematical foundations and presenting empirical analysis, demonstrating its advancement in computational mathematics. Compared to conventional multi-grid and parallel computing techniques, we assess QMGRIT's computing capabilities and efficiency; e.g., on QMGRIT's forerunner algorithm, MGRIT, we find a notable improvement with QMGRIT. This thesis also presents a parallelization paradigm for computing performance improvement and time-energy-saving in high-performance computing (HPC) environments: ghosted QMGRIT (gQMGRIT). To improve convergence rates and resource allocation by a range of scenarios, we evaluate the gQMGRIT and show the effectiveness of the paradigm for QMGRIT solutions. In general, the work seeks to benefit the scientific computing community by offering insight into the design of an innovative combination of two: a parallel-in-time multi-grid algorithm that supplies the ghosted approach of parallelization for the periodic problems; and the objective is to facilitate the more effective treatment of challenging, time-periodic PDEs with impacts that go beyond academia and encompass diverse scientific and engineering fields.
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