Analysis of the second order accurate uniform equilibrium flux method and its graphics processing unit acceleration

Abstract

The extension of the Uniform Equilibrium Flux Method (UEFM) to second order accuracy in space is presented. This extension is made possible through the recasting of the original UEFM flux expressions from a volumetric flux to a surface flux, allowing for reconstruction through a Taylor series expansion of the resulting split surface fluxes at the cell interfaces. By doing so, we avoid the difficulties associated with integration of the gradient terms over velocity and physical space as required by the original UEFM fluxes. Analysis of the dissipative qualities of the renewed direction split UEFM flux expressions demonstrate that the numerical dissipation is a function of Mach number, with increasing amounts of dissipation present with increasing Mach numbers. Following this analysis, the higher order UEFM fluxes are applied to large scale parallel computation using Graphics Processing Units, or GPUs, through the use of CUDA. The vector split nature of the UEFM fluxes are well suited to GPU computation due to the high degree of locality. This parallelization is performed using a cell-based parallel paradigm through the creation of several key CUDA kernels for the calculation of split fluxes, gradient of split fluxes and state related computations. The algorithm is executed entirely on the GPU device, with the host remaining idle during the computation stage. The GPU accelerated UEFM algorithm is then applied to the solution of several two dimensional benchmark problems. Speedup of approximately 200 and 171 times for first order accuracy and second order accuracy respectively is demonstrated when using an Nvidia Tesla C2075 computing GPU compared to that of a single core of an Intel Xeon E5-2760 CPU.