A New Software and Hardware Parallelized Floating Random-Walk Algorithm for the Modified Helmholtz Equation Subject to Neumann and Mixed Boundary Conditions

Abstract

A new floating random-walk algorithm for the one-dimensional modified Helmholtz equation subject to Neumann and mixed boundary conditions problems is developed in this paper. Traditional floating random-walk algorithms for Neumann and mixed boundary condition problems have involved “reflecting boundaries” resulting in relatively large computational times. In a recent paper, we proposed the elimination of the use of reflecting boundaries through the use of novel Green’s functions that mimic the boundary conditions of the problem of interest. The methodology was validated by a solution of the one-dimensional Laplace’s equation. In this paper, we extend the methodology to the floating random-walk solution of the onedimensional modified Helmholtz equation, and excellent agreement has been obtained between an analytical solution and floating random-walk results. The algorithm has been parallelized and a near linear rate of parallelization has been obtained with as many as thirty-two processors. These results have previously been published in [1]. In addition, a GPU implementation employing 4096 simultaneous threads displayed a similar near-linear parallelization gain and a one to two orders of magnitude improvement over the CPU implementation. An immediate application of this research is in the numerical solution of the electromagnetic diffusion equation in magnetically permeable and electrically conducting objects with applications in dielectrometry and magnetometry sensors that have the ability to detect sub-surface objects such as landmines. The ultimate goal of this research is the application of this methodology to the solution of aerodynamical flow problems.