Group Explicit Iterative Method Research Articles

The Alternating Group Explicit Iterative Method for the Regularized Long-Wave Equation

The Alternating Group Explicit Iterative Method for the Regularized Long-Wave Equation

An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions

<abstract><p>In this paper, a new modified hybrid explicit group (MHEG) iterative method is presented for the efficient and accurate numerical solution of a time-fractional diffusion equation in two space dimensions. The time fractional derivative is defined in the Caputo sense. In the proposed method, a Laplace transformation is used in the temporal domain, and, for the spatial discretization, a new finite difference scheme based on grouping strategy is considered. The unique solvability, unconditional stability and convergence are thoroughly proved by the matrix analysis method. Comparison of numerical results with analytical and other approximate solutions indicates the viability and efficiency of the proposed algorithm.</p></abstract>

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.

New High Accuracy Non-polynomial Spline Group Explicit Iterative Method for Two Dimensional Elliptic Boundary Value Problems

In this paper, we propose a new high accuracy method based on non-polynomial spline for the numerical solution of two-dimensional elliptic partial di erential equations. Using a non-polynomial spline approximation in the xdirection and central di erence in the y- direction, we obtain a new nine-point compact finite di erence formulation. A four point group explicit (GE) iterative scheme with an acceleration tool is then applied to the obtained system. The formulation procedure is presented in detail. The eciency of the proposed method is then illustrated by some test problems. The numerical results are found to be in good agreement with the exact solutions

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.

New explicit group iterative methods in the solution of three dimensional hyperbolic telegraph equations

In this paper, new group iterative numerical schemes based on the centred and rotated (skewed) seven-point finite difference discretisations are proposed for the solution of a three dimensional second order hyperbolic telegraph equation, subject to specific initial and Dirichlet boundary conditions. Both schemes are shown to be of second order accuracies and unconditionally stable. The scheme derived from the rotated grid stencil results in a reduced linear system with lower computational complexity compared to the scheme derived from the centred approximation formula. A comparative study with other common point iterative methods based on the seven-point centred difference approximation together with their computational complexity analyses is also presented.

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

Coupled Reduced Alternating Group Explicit Algorithm for Third Order Cubic Spline Method on a Non-uniform Mesh for Nonlinear Singular Two Point Boundary Value Problems

In this paper, we discuss the two parameter coupled reduced alternating group explicit (CRAGE) and Newton-CRAGE iteration methods to obtain the solution of viscous Burgers’ equation using a third order cubic spline numerical method, on non-uniform mesh. The proposed iterative methods are applicable to both singular and non-singular problems. The convergence of the CRAGE iteration method is discussed in detail. We have compared the results of proposed CRAGE iteration method with the results of corresponding two parameter alternating group explicit iterative method to demonstrate computationally the efficiency of the proposed method.

A Single Sweep AGE Algorithm based on Off-Step Discretization for the Solution of Viscous Burgers’ Equation on a Variable Mesh

In this paper, we discuss a new single sweep alternating group explicit iteration method, along with a third order numerical method based on off-step discretization on a variable mesh to solve nonlinear viscous Burgers’ equation subject to given natural boundary conditions. The proposed method is also applicable to both singular and non-singular problems, which is main attraction of our work. The convergence of the proposed method is discussed in detail. We compared the results of proposed iteration method with the results of corresponding double sweep alternating group explicit iteration method to demonstrate computationally the efficiency of the proposed method.

A new modified group explicit iterative method for the numerical solution of time dependent viscous Burgers' equation

In this article, we discuss a new smart alternating group explicit method based on off-step discretization for the solution of time dependent viscous Burgers' equation in rectangular coordinates. The convergence analysis for the new iteration method is discussed in details. We compared the results of Burgers' equation obtained by using the proposed iterative method with the results obtained by other iterative methods to demonstrate computationally the efficiency of the proposed method.

A Single Sweep AGE Algorithm on a Variable Mesh Based on Off-Step Discretization for the Solution of Nonlinear Burgers’ Equation

We discuss a new single sweep alternating group explicit iteration method, along with a third-order numerical method based on off-step discretization on a variable mesh to solve the nonlinear ordinary differential equation y′′=f(x,y,y′) subject to given natural boundary conditions. Using the proposed method, we have solved Burgers’ equation both in singular and nonsingular cases, which is the main attraction of our work. The convergence of the proposed method is discussed in detail. We compared the results of the proposed iteration method with the results of the corresponding double sweep alternating group explicit iteration methods to demonstrate computationally the efficiency of the proposed method.

New explicit group iterative methods in the solution of two dimensional hyperbolic equations

In this paper, we present the development of new explicit group relaxation methods which solve the two dimensional second order hyperbolic telegraph equation subject to specific initial and Dirichlet boundary conditions. The explicit group methods use small fixed group formulations derived from a combination of the rotated five-point finite difference approximation together with the centered five-point centered difference approximation on different grid spacings. The resulting schemes involve three levels finite difference approximations with second order accuracies. Analyses are presented to confirm the unconditional stability of the difference schemes. Numerical experimentations are also conducted to compare the new methods with some existing schemes.

A Class of High-precision Finite Difference Parallel Algorithms for Convection Equations

Convection equation is an important type of mathematical model to describe a variety of physical phenomena. The research on its numerical analysis methods has been the focus of many researchers. Evans [1] firstly proposed the group explicit method for solving the convection equation. Other researchers, including Jingliang Chen [2], Jinfu Lu [3], and Chengri Jin [4] etc., then further improved Evans’s work. Most existing group explicit methods cannot achieve the high accuracy and are limited to second order accuracy. Based on alternating group explicit method’s idea, Evans [5] proposed an alternating group explicit iteration method for solving the parabolic equation. This method not only has good absolute stability and accuracy, it is also suitable for parallel computing. Based on this method, Bolin Zhang [6], Rohallah Tavakoli [7] has proposed several algorithms which could achieve absolute stability and high accuracy. This works integrates the grouping explicit method with numerical boundary conditions and proposes an alternating group explicit iteration method with high accuracy for solving the initial-boundary value problem of convection equations. Our method’s accuracy reaches to four order. Experimental results show the promising performance of our method compared to other existing methods.

Approximation of Iteration Number for Gauss-Seidel Using Redlich-Kister Polynomial

Problem statement: Development of mathematical models based on set of observed data plays a crucial role to describe and predict any phenomena in science, engineering and economics. Therefore, the main purpose of this study was to compare the efficiency of Arithmetic Mean (AM), Geometric Mean (GM) and Explicit Group (EG) iterative methods to solve system of linear equations via estimation of unknown parameters in linear models. Approach: The system of linear equations for linear models generated by using least square method based on (m+1) set of observed data for number of Gauss-Seidel iteration from various grid sizes. Actually there were two types of linear models considered such as piece-wise linear polynomial and piece-wise Redlich-Kister polynomial. All unknown parameters of these models estimated and calculated by using three proposed iterative methods. Results: Thorough several implementations of numerical experiments, the accuracy for formulations of two proposed models had shown that the use of the third-order Redlich-Kister polynomial has high accuracy compared to linear polynomial case. Conclusion: The efficiency of AM and GM iterative methods based on the Redlich-Kister polynomial is superior as compared to EG iterative method.

Application of the alternating group explicit method for convection-diffusion equations

Based on an unconditionally stable finite difference implicit scheme, we present a concept of deriving a class of effective alternating group explicit iterative method for periodic boundary value p...

On the solution of Poisson equation by alternating group explicit iterative method

In this paper the alternating group explicit (AGE) iterative methods which are modified analogues of the classical alternating direction implicit (ADI) methods, are introduced. Poisson equation is transformed a linear system using suitably finite difference equations and also the solution of this system is done with AGE methods. Numerical extraordinary results are obtained by using Maple V computer programming.

The spline alternating group explicit (splage) method for two dimensional problems

A new group explicit iterative method based on cubic spline approximations is presented for the numerical solution of partial differential equations. The numerical results obtained confirm the viability of the method.

Numerical simulation of natural convection in a square cavity using vorticity-velocity formulation.

Two-dimensional incompressible Navier-Stokes equations in vorticity-velocity formulation and the energy equation for temperature have been solved using a method of lines approach. The spatial derivatives are discretized by means of the central finite differential approximation, and the resulting system of the ordinary differential equations in time is integrated by a rational Runge-Kutta (RRK)scheme. Poisson equations for velocity are solved by a Group Explicit Iterative (GEI) method, using the multigrid technique. The GEI method is a fully explicit scheme so it takes advantage of the supercomputer's architecture. As the verification of the scheme, the classical problem of natural convection in a square cavity of Boussinesq fluid has chosen. Numerical calculations are obtained at a Prandtle number of 0.71 corresponding to air and Rayleigh number up to 108. The results of calculation essentially agree well with those of de Vahl Davis.

Numerical solutions to the incompressible Navier-Stokes equations in vorticity-velocity form by using the group explicit iterative method.

A numerical method for computing two-dimensional incompressible flows is presented. The steady vorticcity-velocity equations are discretized with the central finite difference approximation and the resulting equations are solved by using the Multi-Grid method with the Group-Explicit-Iterative method. The driven flow in a square cavity is used as a test problem. Calculations are carried out for the case with Reynolds number as high as 3200 on 257×257 grid points. The computed results are in good agreement with those based on the vorticity-stream function equations, and it is found that large residuals of the numerical solutions for the equation of continuity and the defining equation of vorticity occur near the corners on the moving surface.

Numerical analysis for square cavity flow at high Reynolds number using GEI method.

A new iterative method has been developed for calculating steady flows in a square cavity. The steady stream function vorticity equations are discretized with the central finite difference approximation and the resulting system of equations are solved by using the group-explicit-iterative (GEI) method. The calculations are carried out for the case with Reynolds number as high as 10000 on 257×257 grid points. The computed results are in excellent agreement with those by other methods.