The Higher Accuracy Fourth-Order IADE Algorithm

Abstract

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.

Solution of a One-Dimension Heat Equation Using Higher-Order Finite Difference Methods and Their Stability

“…This complexity was incorporated in different ways such as Muhiddin and Sulaiman [1] incorporated that using lower-order scheme, Paul and Ali [4] suggested maintaining boundary temperature at all grid points outside the boundary, Paul and Ali [7] used weighted average technique, and following Hicks and…”

“…It is often solved using different analytical and numerical tools [1]- [6]. For example, Gorguis and Chan [3] solved (1 + 1) dimensional heat equation analytically using Adomian decomposition method and compared obtained solution with that came through separation of variable method; Paul and Ali [4] and Ahmad and Yaacob [6] used method of lines in coordination with higher-order finite difference approximation to solve the equation; Muhiddin and Sulaiman [2] solved the equation using fourth-order Crank-Nicolson (CN4) finite difference method (FDM) and fourth-order standard implicit FDM (BTCS) and made a comparison of obtained results; fourth-order iterative alternating decomposition explicit method of Mitchell and Fairweather was exercised in the study due to Mansor et al [4]. As we cannot find the analytical solution of most of the PDEs, efficient and faster numerical techniques are highly appreciable in research community for solving those.…”

A Distributed Memory Parallel Fourth-Order IADEMF Algorithm

“…A recent study made by Mansor [11] involved the development of a convergent and unconditionally stable fourth-order IADEMF sequential algorithm (IADEMF4). The proposed scheme is found to be capable of enhancing the accuracy of the original corresponding method of the secondorder, that is, the IADEMF2.…”

“…In this section, the development of the IADEMF4 algorithm [11] is briefly reviewed. Consider the onedimensional heat equation 1which models the flow of heat in a homogeneous unchanging medium of finite extent, in the absence of heat source.…”