We consider three level difference replacements of parabolic equations focusing on the heat equation in two space dimensions. Through a judicious splitting of the approximation, the scheme qualifies as an alternating direction implicit (ADI) method. Using the well known fact of the parabolic-elliptic correspondence, we shall derive a two stage iterative procedure employing a fractional splitting strategy applied alternately at each intermediate time step to the one dimensional heat equation. As the basis of derivation is the unconditionally stable (4,2) accurate ADI scheme, this method is convergent, computationally stable and highly accurate.
This study develops the novel fourth-order iterative alternating decomposition explicit (IADE) method of Mitchell and Fairweather (IADEMF4) algorithm for the solution of the one-dimensional linear heat equation with Dirichlet boundary conditions. The higher-order finite difference scheme is developed by representing the spatial derivative in the heat equation with the fourth-order finite difference Crank-Nicolson approximation. This leads to the formation of pentadiagonal matrices in the systems of linear equations. The algorithm also employs the higher accuracy of the Mitchell and Fairweather variant. Despite the scheme’s higher computational complexity, experimental results show that it is not only capable of enhancing the accuracy of the original corresponding method of second-order (IADEMF2), but its solutions are also in very much agreement with the exact solutions. Besides, it is unconditionally stable and has proven to be convergent. The IADEMF4 is also found to be more accurate, more efficient, and has better rate of convergence than the benchmarked fourth-order classical iterative methods, namely, the Jacobi (JAC4), the Gauss-Seidel (GS4), and the successive over-relaxation (SOR4) methods.
The fourth-order finite difference Iterative Alternating Decomposition Explicit Method of Mitchell and Fairweather (IADEMF4) sequential algorithm has demonstrated its ability to perform with high accuracy and efficiency for the solution of a one-dimensional heat equation with Dirichlet boundary conditions. This paper develops the parallelization of the IADEMF4, by applying the Red-Black (RB) ordering technique. The proposed IADEMF4-RB is implemented on multiprocessor distributed memory architecture based on Parallel Virtual Machine (PVM) environment with Linux operating system. Numerical results show that the IADEMF4-RB accelerates the convergence rate and largely improves the serial time of the IADEMF4. In terms of parallel performance evaluations, the IADEMF4-RB significantly outperforms its counterpart of the second-order (IADEMF2-RB), as well as the benchmarked fourth-order classical iterative RB methods, namely, the Gauss-Seidel (GS4-RB) and the Successive Overrelaxation (SOR4-RB) methods.
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