Group Explicit Method Research Articles

Fractional Newton Explicit Group Method for Time-Fractional Nonlinear Porous Medium Equations

Fractional Newton Explicit Group Method for Time-Fractional Nonlinear Porous Medium Equations

Numerical Approximation to Porous Medium Equation Using a Quarter-Sweep Based Finite Difference and Explicit Four-Points Group

In this study, numerical approximation is used to tackle the one-dimensional nonlinear porous media problem. The emphasis is on using a quarter-sweep finite difference technique to apply the approximation. The quarter-sweep computation approach is highly helpful in terms of reducing complexity while solving big and sparse linear equations. This is done in order to produce a finite difference scheme that works well for the porous medium equation. Internal calculations are carried out using an explicit four-points group methodology, and the resulting nonlinear system is solved using the Newton method. To demonstrate the efficiency of the suggested finite difference approximation and the explicit four-points group technique together, several examples involving the porous medium equation are described. Analyzing the maximum number enables an assessment of the computational time and iteration efficiency of the numerical approach. The method's accuracy is further evaluated by looking at the largest absolute errors the grid points can produce. The suggested method is compared with a half-sweep finite difference method that uses the Newton explicit group method and a conventional quarter-sweep finite difference approximation.

Numerical Solution of Two-Dimensional Time Fractional Mobile/Immobile Equation Using Explicit Group Methods.

In this paper, we shall present the development of two explicit group schemes, namely, fractional explicit group (FEG) and modified fractional explicit group (MFEG) methods for solving the time fractional mobile/immobile equation in two space dimensions. The presented methods are formulated based on two Crank-Nicolson (C-N) finite difference schemes established at two different grid spacings. The stability and convergence of order O(tau ^{2-alpha }+h^2) are rigorously proven using Fourier analysis. Several numerical experiments are conducted to verify the efficiency of the proposed methods. Meanwhile, numerical results show that the FEG and MFEG algorithms are able to reduce the computational times and iterations effectively while preserving good accuracy in comparison to the C-N finite difference method.

Fast hybrid explicit group methods for solving 2D fractional advection-diffusion equation

<abstract><p>In recent years, fractional partial differential equations (FPDEs) have been viewed as powerful mathematical tools for describing ample phenomena in various scientific disciplines and have been extensively researched. In this article, the hybrid explicit group (HEG) method and the modified hybrid explicit group (MHEG) method are proposed to solve the 2D advection-diffusion problem involving fractional-order derivative of Caputo-type in the temporal direction. The considered problem models transport processes occurring in real-world complex systems. The hybrid grouping methods are developed based upon a Laplace transformation technique with a pair of explicit group finite difference approximations constructed on different grid spacings. The proposed methods are beneficial in reducing the computational burden resulting from the nonlocality of fractional-order differential operator. The theoretical investigation of stability and convergence properties is conducted by utilizing the matrix norm analysis. The improved performance of the proposed methods against a recent competitive method in terms of central processing unit (CPU) time, iterations number and computational cost is illustrated by several numerical experiments.</p></abstract>

A New Technique to Speed Up Group Methods for Solving Hyperbolic Telegraph Equations

Numerous methods have been introduced in the literature for numerical solution of two-dimensional hyperbolic differential equations (telegraph equations). Improved techniques using explicit group methods derived from the standard and skewed (rotated) finite difference operators have been developed over the last few years in solving the linear systems that arise from the discretization of the several types of partial differential equations. In this paper, we present a preliminary study of the formulation of new preconditioned scheme based on explicit group relaxationmethods for the difference solution of the telegraph equations. The efficient and robustness of these new formulations over the existing explicit group schemes demonstrated through numerical experiments.

Group Splitting with SOR/AOR Methods for Solving Boundary Value Problems: A Computational Comparison

Many researchers are working on the explicit group methods as the alternative methods for solving several boundary value problems. These methods have been shown to be much faster than the other point iterative methods in solving the elliptic partial differential equations (EPDEs), which is due to the formers’ overall lower computational complexities. This paper is concerned with the application of a suitable Explicit Group (EG) iterative method for solving EPDEs. This study will compare several iterative methods such that S5-point-SOR,4 Point-EGSOR, 5S-point-AOR, and 4 Point-EGAOR. Numerical experiments were carried out to confirm our results by using MATLAB software. The results reveal that 4 Point-EGAOR is the most superior method among these methods.

Numerical Solution for Fuzzy Diffusion Problem via Two Parameter Alternating Group Explicit Technique

The computational technique has become a significant area of study in physics and engineering. The first method to evaluate the problems numerically was a finite difference. In 2002, a computational approach, an explicit finite difference technique, was used to overcome the fuzzy partial differential equation (FPDE) based on the Seikkala derivative. The application of the iterative technique, in particular the Two Parameter Alternating Group Explicit (TAGE) method, is employed to resolve the finite difference approximation resulting after the fuzzy heat equation is investigated in this article. This article broadens the use of the TAGE iterative technique to solve fuzzy problems due to the reliability of the approaches. The development and execution of the TAGE technique towards the full-sweep (FS) and half-sweep (HS) techniques are also presented. The idea of using the HS scheme is to reduce the computational complexity of the iterative methods by nearly/more than half. Additionally, numerical outcomes from the solution of two experimental problems are included and compared with the Alternating Group Explicit (AGE) approaches to clarify their feasibility. In conclusion, the families of the TAGE technique have been used to overcome the linear system structure through a one-dimensional fuzzy diffusion (1D-FD) discretization using a finite difference scheme. The findings suggest that the HSTAGE approach is surpassing in terms of iteration counts, time taken, and Hausdorff distance relative to the FSTAGE and AGE approaches. It demonstrates that the number of iterations for HSTAGE approach has decreased by approximately 71.60-72.95%, whereas for the execution time, the implementation of HSTAGE method is between 74.05-86.42% better. Since TAGE is ideal for concurrent processing, this method has been seen as the key benefit as it consumes sets of independent tasks that can be performed at the same time. The ability of the suggested technique is projected to be useful for the advanced exploration in solving any multi-dimensional FPDEs.

A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh\u2013Stokes problem for a heated generalized second-grade fluid

In this article, a new explicit group iterative scheme is developed for the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. The proposed scheme is based on the high-order compact Crank–Nicolson finite difference method. The resulting scheme consists of three-level finite difference approximations. The stability and convergence of the proposed method are studied using the matrix energy method. Finally, some numerical examples are provided to show the accuracy of the proposed method.

On a new weight tri-diagonal iterative method and its applications

In this article, a new weight tri-diagonal iterative method in solving system of linear equation is proposed and its convergence is discussed. The set of methods in solving linear system generated by high-order scheme for solving one-dimensional Poisson equation and one-dimensional heat equation with periodic boundary are presented. The numerical experiments shows that the proposed method demonstrates a better performance compared with weight Jacobi, successive-over relaxation and alternating group explicit methods.

A new sixth-order approximation for nonlinear two-point boundary value problems: application of single-step alternating group explicit iteration method to engineering problems

We report a new three-point compact sixth-order approximation for the solution of nonlinear two-point boundary value problems $$ {-}w^{\prime\prime} + f\left( {t,w} \right) = 0 $$, subject to natural boundary conditions prescribed at two end points. We also discuss the application of single-step alternating group explicit (SAGE) iteration method to the nonlinear difference equation as a computational tool. The error analysis of the SAGE iteration method is discussed briefly. We have compared the results obtained by using the proposed SAGE iteration method with the results obtained by corresponding two-step alternating group explicit iteration method to demonstrate experimentally the superiority of the proposed method.

Alternating Asymmetric Iterative Algorithm Based on Domain Decomposition for 3D Poisson Problem

Poisson equation is a widely used partial differential equation. It is very important to study its numerical solution. Based on the strategy of domain decomposition, the alternating asymmetric iterative algorithm for 3D Poisson equation is provided. The solution domain is divided into several sub-domains, and eight asymmetric iterative schemes with the relaxation factor for 3D Poisson equation are constructed. When the numbers of iteration are odd or even, the computational process of the presented iterative algorithm are proposed respectively. In the calculation of the inner interfaces, the group explicit method is used, which makes the algorithm to be performed fast and in parallel, and avoids the difficulty of solving large-scale linear equations. Furthermore, the convergence of the algorithm is analyzed theoretically. Finally, by comparing with the numerical experimental results of Jacobi and Gauss Seidel iterative algorithms, it is shown that the alternating asymmetric iterative algorithm based on domain decomposition has shorter computation time, fewer iteration numbers and good parallelism.

Saul’yev and Group Explicit Methods

The Saul’yev methods for parabolic equations are implicit in form, but can be solved explicitly and are therefore interesting in connection with non-linear problems. Abdullah’s Group Explicit methods are parallel in nature and therefore interesting when using parallel computers. The main objective of this paper is to study the accuracy of these methods. Using global error estimation we show that for all these methods the time step must be bounded by the square of the space step size to ensure a global error which can be estimated. As a curiosity we show that the two original Saul’yev methods in fact solve two different differential equations.

Application of Alternate Group Explicit Method for Linear Boundary Value Problems

Application of Alternate Group Explicit Method for Linear Boundary Value Problems

求解抛物方程的MPI并行方法

数值求解偏微分方程广泛应用于数学与工程领域。大规模数值计算在当今科学技术运用中得到飞速发展，其中可并行的有限差分格式受到越来越多的重视。在本文中，主要阐述了经典的分组显示方法求解抛物方程，并简单扼要的分析了该格式的建立以及稳定性。随后本文着重介绍了如何在MPI并行环境下对该格式进行数值计算，构建了两种不同的并行计算模型，即阻塞通信(等待模式)和非阻塞通信(即非等待模式)模式。并与非并行状态下的差分格式做出比较，结果表明，相对于一个进程求解偏微分方程，两种模式都表现出较好的效果，而且非阻塞通信相较于阻塞通信模式亦表现出较好的并行效率。 Many applications in mathematics and engineering involve numerical solutions of partial diffe-rential equations (PDEs). The demands of large-scale computing are quickly increasing in modern science and technology, and parallel computing has received more and more attention. In this paper, the main idea is that classical Group Explicit method (GEM) for parabolic equations, the group explicit method is established briefly and the stability analysis of the method is indicated simply. Then we focus on how to calculate the format in MPI parallel environment. Two parallel MPI algorithms are established and compared with non-parallel algorithm based on GEM. They are MPI block communication (wait communication) and non-blocking communication (no-wait communication). These two MPI schemas both better than one single process to calculate numerical solutions use group explicit method. Also, the non-blocking communication program has higher computational efficiency than blocking communication program.

Half-sweep two parameter alternating group explicit iterative method applied to fuzzy Poisson equation

Iterative methods particularly the Two Parameter Alternating Group Explicit (TAGE) methods are used to solve system of linear equations generated from the discretization of two-point fuzzy boundary value problems (FBVPs). The formulation and implementation of the Full-Sweep TAGE (FSTAGE) and Half-Sweep TAGE (HSTAGE) methods are also presented. Then numerical experiments are carried out onto two example problems to verify the effectiveness of the method. The results show that TAGE method is superior compare to AGE method in the aspect of number of iterations, execution time and Hausdorff distance.

A new compact alternating group explicit iteration method for the solution of nonlinear time-dependent viscous Burgers’ equation

We discuss a new single-sweep compact alternating group explicit method for solving the time-dependent viscous Burgers’ equation both in Cartesian and polar coordinates. An error analysis for the new iterative method is discussed in detail.We compared the results of the proposed iterative method with the results of a corresponding double-sweep alternating group explicit (AGE) iterative method to demonstrate computationally the efficiency of the proposed method.

A new coupled reduced alternating group explicit method for nonlinear singular two-point boundary value problems on a variable mesh

In this paper, we discuss a new coupled reduced alternating group explicit (CRAGE) and Newton-CRAGE iteration methods to solve the nonlinear singular two-point boundary value problems u″ = f(r, u, u′), 0 < r < 1 subject to given natural boundary conditions u(0) = A1, u(1) = A2 where A1 and A2 are finite constants, along with a third-order numerical method on a geometric mesh. The proposed method is applicable to singular and nonsingular problems. We have discussed the convergence of the CRAGE iteration method in detail. The results obtained from the proposed CRAGE iteration method are compared with the results of the corresponding two-parameter alternating group explicit (TAGE) iteration methods to demonstrate computationally the efficiency of the proposed method.

Новый компактный явный групповой итерационный метод переменных направлений с одной прогонкой для решения нелинейного, зависящего от времени, вязкого уравнения Бюргерса

Новый компактный явный групповой итерационный метод переменных направлений с одной прогонкой для решения нелинейного, зависящего от времени, вязкого уравнения Бюргерса

Implementation of TAGE Method Using Seikkala Derivatives Applied to Two-Point Fuzzy Boundary Value Problems

Iterative methods particularly the Two-Parameter Alternating Group Explicit (TAGE) methods are used to solve system of linear equations generated from the discretization of two-point fuzzy boundary value problems (FBVPs). The formulation and implementation of the TAGE method are also presented. Then numerical experiments are carried out onto two example problems to verify the effectiveness of the method. The results show that TAGE method is superior compared to GS method in the aspect of number of iterations, execution time, and Hausdorff distance.

Alternating Group Explicit Iterative Methods for One-Dimensional Advection-Diffusion Equation

The finite difference method such as alternating group iterative methods is useful in numerical method for evolutionary equations and this is the standard approach taken in this paper. Alternating group explicit (AGE) iterative methods for one-dimensional convection diffusion equations problems are given. The stability and convergence are analyzed by the linear method. Numerical results of the model problem are taken. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show that the behavior of the method with emphasis on treatment of boundary conditions is valuable.