Parallel Numerical Solution of Partial Differential Equations

Abstract

Simulation of scientific problems is an important aspect of natural and engineering sciences. Simulations demanding higher accuracy or involving larger data sets require higher computing power. Complex mathematical models involving partial differential equations (PDEs) from computational fluid dynamics (CFD) are some examples of these simulations. The conventional serial computers are not able to meet the increasing demand of computation power for such applications and the only rescue is parallel or high-performance computing. This study presents research regarding parallel numerical solution of PDEs. Message Passing Interface (MPI) clusters and Graphic Processor Units (GPUs) being the leading platforms for parallel computing were used for simulation of results. The research begins with the unified analysis of the existing parallel iterative algorithms using MPI. A set of diverse PDEs was solved using the MPI cluster. After getting an insight of iterative methods for MPI platform, the parallel system with shared memory architecture was experimented. The most advent platform in this regard is GPUs having thousands of concurrent running cores along with many Giga bytes (GBs) of memory. 3D Laplace equation was solved using twelve different kernels to exploit the memory hierarchy of GPU and an efficient technique involving surfacing pointer’s capability of GPU was materialized. The GPU kernel exhibiting said features gained a speedup of 70 as compared to serial version of same program running on Intel core i5 processor. The derived technique was further extended to simulate the compressible, high-speed flows modeled by Navier Stokes equations using GPU. Four different structured geometries were modeled; the governing equations were solved using modified RK4 method and TVD scheme was used for shockwave capturing. The derived technique was also used to simulate the flow in micro channel using Lattice Boltzmann Method. The GPU results show a speedup of 23 and 77 as compared with serial variants of codes running on conventional core i5®CPU for both cases respectively. It is evident from obtained results that the performance of CFD and other compute intensive application can be enhanced many folds by using the devised technique involving surface pointers in GPU computation.