Keywords: local numerical method, meshless, OpenMP, GPU, parallel.

A simple and intuitive numerical methodology for solving systems of non-linear partial differential equations (PDEs) in parallel is presented in this paper. The solution algorithms form a general approach towards the evaluation of an arbitrary differential operator based on a local approximation principle. Besides the simpler formulation, the local solution procedure also enables higher parallel efficiency. From the computation point of view, the localization reduces inter-processor communication, which is often a bottleneck of parallel algorithms. The presented numerical approach, also referred to as a meshless local strong form method (MLSM), does not require any kind of special topological relations between computational points and relies only on searching the closest neighbouring points, therefore it can be also used to consider complex domains and problems with moving boundaries. The MLSM can be also easily upgraded or altered to treat anomalies such as sharp discontinues or other obscure situations, which might occur in complex simulations. In this paper the MLSM is demonstrated on three different problems governed by systems of PDEs, namely the simulation of semiconductors, a natural convection problem and the simulation of binary solidification. The parallel efficiency of the MLSM implementation is demonstrated through the speedup measurements on multicore and multi GPU computer architectures. The multicore speedup results are also supported by low-level CPU cache utilization measurements.

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