Analysis Of Parallel Algorithms For Time-Dependent Three Dimentional Heat Partial Differential Equation Based On PETSc

The three dimensional heat equation, a partial differential equation, has been used to solve time dependent methods in a various number of fields including engineering, some of which include groundwater flow and reservoir simulation. The equation has been used to solve problems that where too of complex, too large and required numerous amounts of processing power. In this paper, a parallel-computing strategy is presented to accelerate the simulation process and to try to identify improvements in making full utilization of computer resources as to reach peak machine capabilities. The distributed memory and compressed matrices technology has been adopted for both the process of storage and evaluation of large-scale sparse matrices.Krylov subspace methods and preconditioners have been introduced to assemble and solve the linear systems of equations. Comparisons where made between two algorithms, namely the (Conjugate Gradient method (CG)) and (Generalized Minimal Residual GMRES)) algorithms because they are widely recognized as the standard methods for the solution of symmetric positive definite systems with different preconditioners. The preconditoners used in the numerical experiments are the Block Jacobi (BJacobi), Additive Schwarz Method (ASM) and MultiGrid (GM)) as they can be used to speed up the convergence rate of iterative methods. The code implementation of this strategy is written in high-level abstractions based on object-oriented technology which promotes code reuse, flexibility and helps to decouple issues of parallelism from algorithm choices. Experiments based on the three-dimensional heat equation have been carried out on Linux clusters using the Portable Extensible Toolkit for Scientific computation and the results demonstrate that this strategy has achieved desirable speedup and efficiency. Numerical experiments were carried out on different grid sizes of 50*50*50, 100*100*50 and 100*100*100. The results reveal that a combination of the CG algorithm and Block Jacobi preconditioner provide the best parallel solution compared to other as it was more suitable at solving the three-dimensional heat equation. With this combination, the parallel program had the most ideal speedup and efficiency.