Numerical Evaluation of Quarter-Sweep KSOR Method to Solve Time-Fractional Parabolic Equations

Abstract

We study the performance of the combination of quarter-sweep iteration concept with the Kaudd Successive Over-Relaxation (KSOR) iterative method in solving the discretized one-dimensional time-fractional parabolic equation. We called the mixed of these two concepts as QSKSOR. The time-fractional derivative in Grünwald sense, together with the implicit finite difference scheme was used to discretized the tested problems to form the quarter-sweep implicit finite difference approximation equations in the sense of Grünwald type. This approximation equation of half-sweep will then generate a linear system. Next, we used the proposed QSKSOR iterative method to the generated linear systems before comparing the effectiveness between the other family of KSOR method, FSKSOR and HSKSOR with respect to the full-and half-sweep cases respectively. To do so, three examples are included. The results of this study show the superiority of the QSKSOR iterative method in terms of iteration numbers and execution time in comparison to the other two methods.

Quarter-Sweep Preconditioned Relaxation Method, Algorithm and Efficiency Analysis for Fractional Mathematical Equation

“…For the numerical experiment, two problems of the time-fractional diffusion equation were selected to test the performance of the QSPSOR. For the efficiency comparison and analysis, two existing methods from our previous work were used, abbreviated as FSP-SOR [19] and HSPSOR [20]. To compare the performance of these methods, three criteria were considered, namely, η-representing the number of iterations, sec.-representing the execution time of the C++ simulation code, and ε-representing the magnitude of the absolute error.…”

“…A better understanding of FDE models can be achieved using an efficient numerical method called the quarter-sweep preconditioned relaxation method. It should be noted that this paper extends the works from [19] and [20], which implemented the standard implicit FDM with preconditioned relaxation and the half-sweep difference scheme with preconditioned relaxation, respectively, for solving the time FDE. The work in [19] showed the improvement in the number of iterations and execution time after implementing a preconditioned successive over-relaxation with implicit FDM using the Caputo approach to solve the time FDE.…”

“…It should be noted that this paper extends the works from [19] and [20], which implemented the standard implicit FDM with preconditioned relaxation and the half-sweep difference scheme with preconditioned relaxation, respectively, for solving the time FDE. The work in [19] showed the improvement in the number of iterations and execution time after implementing a preconditioned successive over-relaxation with implicit FDM using the Caputo approach to solve the time FDE. Another article [20] applied a computational complexity reduction technique called the half-sweep iteration to successfully reduce the computational cost using preconditioned successive over-relaxation, and eventually improve the efficiency of Caputo's finite difference scheme.…”

Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation

“…Several iterative methods for solving a linear system with a large-scale and sparse coefficient matrix are discussed in the literature. The implementations of the point iteration family, such as successive over-relaxation (SOR) [26], [27], accelerated over-relaxation (AOR) [28], [29], and kaudd successive over-relaxation (KSOR) [30], [31], can be used to solve this linear method. Evans [32] invented the explicit group (EG) iteration method for obtaining an approximate solution by grouping the linear system into a sequence of (4x4) linear systems based on the coefficient matrix's characteristics.…”