Fast Iterative Solver for the 2-D Convection-Diffusion Equations

In this paper, we introduce the preconditioned Explicit Decoupled Group (EDG) for solving the two dimensional Convection-Diffusion equation with initial and Dirichlet boundary conditions. The purpose of this paper is to accelerate the convergence rate of the Explicit Decoupled Group (EDG) method by using suitable preconditioned iterative scheme for solving the Convection-Diffusion. The robustness of these new formulations over the existing EDG scheme demonstrated through numerical experiments

Solving 2D Time-Fractional Diffusion Equations by Preconditioned Fractional EDG Method

Fractional differential equations play a significant role in science and technology given that several scientific problems in mathematics, physics, engineering and chemistry can be resolved using fractional partial differential equations in terms of space and/or time fractional derivative. Because of new developments in the analysis and understanding of many complex systems in engineering and sciences, it has been observed that several phenomena are more realistically and accurately described by differential equations of fractional order. Fast computational methods for solving fractional partial differential equations using finite difference schemes derived from skewed (rotated) difference operators have been extensively investigated over the years. The main aim of this paper is to examine a new fractional group iterative method which is called Preconditioned Fractional Explicit Decoupled Group (PFEDG) method in solving 2D time-fractional diffusion equations. Numerical experiments and comparison with other existing methods are given to confirm the superiority of our proposed method

An Adaptive Preconditioner Matrix on N-P Group AOR Iterative Poisson Solver

Hadjidimos [1], proved that the Accelerated OverRelaxation (AOR) is more powerful compared with the other well-known method called the Successive OverRelaxation (SOR) for solving linear systems of equations. The formulation of group iterative schemes for approximating the solution of the two dimensional elliptic partial differential equations have been the subject of intensive study during the last few years. The recent convergence results of nine-point (N-P) group iterative schemes from the Successive OverRelaxation (SOR) family have been presented by Saeed [2]. In this paper, we extend the work of Saeed [2] with the new application of suitable preconditioning techniques to the N-P Group iterative schemes from the Accelerated OverRelaxation (AOR) for solving Poisson’s Equation. The results reveal the significant improvement in number of iterations and execution timings of the proposed preconditioned Group iterative method compared to Preconditioned N-P SOR

Preconditioned Explicit Decoupled Group Methods For Solving Elliptic Partial Differential Equations

Perkembangan yang pesat bagi kaedah beza hingga adalah didorong oleh keperluan untuk mengatasi masalah yang kompleks hari ini dalam sains dan teknologi. The highly concern development of finite difference methods was stimulated by the need to cope with today's complex problems in science and technology

Improved Rotated Finite Difference Method for Solving Fractional Elliptic Partial Differential Equations

Real life problems with fractional partial differential equations (FPDE's) are of great importance, since fractional differential equations accumulate the whole information of the function in a weighted form. This has many applications in physics, chemistry, engineering, etc. For that reason, we need a method for solving such equations, effectively, easy use and applied for different problems. The objective of this paper is to solve fractional elliptic partial differential equations, by using new accelerated version of rotated five point’s approximation method. Experiment results of the test problem are given in order to confirm the superiority of our proposed method

Preconditioned explicit decoupled group methods for solving elliptic partial differential equations.

Perkembangan yang pesat bagi kaedah beza hingga adalah didorong oleh keperluan untuk mengatasi masalah yang kompleks hari ini dalam sains dan teknologi. Keperluan terkini bagi penyelesaian lebih cepat dan untuk menyelesaikan masalah saiz besar yang muncul dalam pelbagai aplikasi dalam bidang sains, seperti pemodelan, simulasi sistem yang besar dan dinamik bendalir. The highly concern development of finite difference methods was stimulated by the need to cope with today’s complex problems in science and technology. The current requirement for faster solutions and for solving large size problems arises in a variety of applications in science, such as modeling, simulation of large systems and fluid dynamics

Preconditioned Modified Explicit Decoupled Group for the Solution of Steady State Navier-Stokes Equation

Combining iterative schemes with suitable preconditioners may improve the rate of the convergence of the methods. However, the real difficulty lies in the construction of the correct preconditioners applied to the formulated schemes. In this paper, the Modified Explicit Decoupled Group Successive Over-Relaxation method is formulated to solve the two dimensional steady-state Navier-Stokes equations. A new block splitting preconditioned matrix is applied to the formulated scheme as an effort to accelerate the convergence rate. Numerical experiments are carried out to confirm the effectiveness of the preconditioner in terms of accuracy and execution timings. Comparison with its unpreconditioned counterpart will also be reported

Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems

The construction of a specific splitting-type preconditioner in block formulation applied to a class of group relaxation iterative methods derived from the centred and rotated (skewed) finite difference approximations has been shown to improve the convergence rates of these methods. In this paper, we present some theoretical convergence analysis on this preconditioner specifically applied to the linear systems resulted from these group iterative schemes in solving an elliptic boundary value problem. We will theoretically show the relationship between the spectral radiuses of the iteration matrices of the preconditioned methods which affects the rate of convergence of these methods. We will also show that the spectral radius of the preconditioned matrices is smaller than that of their unpreconditioned counterparts if the relaxation parameter is in a certain optimum range. Numerical experiments will also be presented to confirm the agreement between the theoretical and the experimental results

Investigating Symmetric Soliton Solutions for the Fractional Coupled Konno–Onno System Using Improved Versions of a Novel Analytical Technique

The present research investigates symmetric soliton solutions for the Fractional Coupled Konno–Onno System (FCKOS) by using two improved versions of an Extended Direct Algebraic Method (EDAM) i.e., modified EDAM (mEDAM) and r+mEDAM. By obtaining precise analytical solutions, this research explores the characteristics and behaviours of symmetric solitons in FCKOS. Further, the amplitude, shape and propagation behaviour of some solitons are visualized by means of a 3D graph. This investigation fosters a more thorough comprehension of non-linear wave phenomena in considered systems and offers helpful insights towards soliton behavior in it. The outcomes reveal that the recommended techniques are successful in constructing symmetric soliton solutions for complex models like the FCKOS