Keywords: OpenMP, superlinear speedup, meshfree, LRBFCM, convective-diffusive problems, fluid flow, de Vahl Davis, natural convection.

The application of the local meshless numerical method (LRBFCM) [1] to the solution of a system of coupled partial differential equations (PDE) is explored. The numerical approach is tested on natural convection based fluid flow problems. The computational domain is spatially discretized using the LRBFCM. Explicit time stepping is used for temporal discretization.The fluid flow part of the solution procedure is coupled locally [2] despite its global nature [3]. Such an approach makes the computations appropriate for implementation on parallel platforms as it does not require global communication. In this paper, the OpenMP based parallelization of the proposed meshless numerical approach is demonstrated. The parallelization performance is explored for the classical de Vahl Davis natural convection case [4]. On two cores, a superlinear speedup of 2.5 is shown in the performance analysis. Through the analysis of computational problem the background of superlinearity is explained. The accumulating L3 caches are identified as a source of such behaviour. The conclusions are supported by measurements of the L3 cache hit rate on the Intel CPU architecture. It is shown that the superlinear speedup occurs when the L3 cache hit rate starts to decline on a single core while it stays slightly above 0.9 on two cores. In this regime, an increased problem size causes the working data set to become too large for the L3 cache of a single core while it still fits into the two L3 caches that are available using two-core system. The single-threaded program slows down compared with the two-threaded program as a result of an increased number of L3 cache accesses.

The usability of the presented meshless numerical framework has been recently demonstrated on highly non-linear and coupled case of the solidification of binary alloy [5], where energy and solute transport govern the double natural convection in a domain filled with porous media and free fluid with moving interphases as well as in the adaptive node distribution strategy [6]. It has been also recently demonstrated that, despite its simplicity, the numerical method presented performs well in comparison to more complicated methods [7].

1

G. Kosec, B. Šarler, "Numerical solution of natural convection problems by a meshless method", Convection and Conduction Heat Transfer, 108-132, 2011. doi:10.5772/23816

2

G. Kosec, B. Šarler, "Solution of thermo-fluid problems by collocation with local pressure correction", International Journal of Numerical Methods for Heat and Fluid Flow, 18, 868-882, 2008. doi:10.1108/09615530810898999

3

J.H. Ferziger, M. Peric, "Computational Methods for Fluid Dynamics", Springer, Berlin, 2002. doi:10.1007/978-3-642-56026-2

4

G. de Vahl Davis, "Natural convection of air in a square cavity: a bench mark numerical solution", International Journal of Numerical Methods in Fluids, 3, 249-264, 1983.

5

G. Kosec, M. Zaloznik, B. Šarler, H. Combeau, "A Meshless Approach Towards Solution of Macrosegregation Phenomena", Computers, Materials & Continua, 580, 1-27, 2011.

6

G. Kosec, B. Šarler, "H-adaptive local radial basis function collocation meshless method", Computers, Materials & Continua, 26, 227-254, 2011.

7

R. Trobec, G. Kosec, M. Šterk, B. Šarler, "Comparison of local weak and strong form meshless methods for 2-D diffusion equation", Engineering Analysis with Boundary Elements, 36, 310-321, 2012. doi:10.1016/j.enganabound.2011.08.009

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