Alternating Group Explicit 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

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.

Numerical simulation of DIC drying process on matlab distributed computing server

<span>Instant controlled pressure drop, also known as DIC, is one of the drying techniques that has been used for texturing, extracting and drying various food products. Mathematical model has been used to explain the drying process, although most of the studies focused on the statistical regression model approach. Due to the limitations of regression model, which neglects the fundamental of dehydration process, t</span><span>his paper presents the development of mathematical models to detect, solve and visualize the three-dimensional (3D) heat and mass transfer in DIC drying process. </span><span>Finite Difference Method (FDM) with central difference formula is used to discretize the mathematical models. A large sparse of system of linear equation (SLE), which represents the actual drying process, is solved by using some numerical methods, such as Jacobi (JB), Red Black Gauss Seidel (RBGS), Alternating Group Explicit with Douglas (AGED) and Brian (AGEB) methods. Based on the numerical results, high execution time and high computational complexity have been shown. In order to reduce the execution time and computational complexity, the parallel algorithm based on domain decomposition technique has been implemented on the MATLAB Distributed Computing Server (MDCS). The parallel algorithm of the numerical methods was evaluated and compared based on the parallel performance metrics: execution time, speed up, efficiency, effectiveness, temporal performance and granularity. From the parallel performance metrics, it was found that the PAGEB approach had better performance, followed by PAGED, PRBGS and PJB methods.</span>

Parallel performance evaluation for curing process of thermoset nanocomposites materials

The investigation towards rate of growth for the curing process of thermoset nanocomposites materials is of great concern to most engineering and chemistry researchers since it is closely related to our natural resources, and has valuable properties. It is important to tailor and control the waste composite materials temperature profile during curing process to reproduce and improve its efficiency. Due to this phenomenon, temperature profile during curing process between two layers of composite materials is considered in this study. The mathematical model of 1D convection–diffusion of the heat equation of thick thermoset composite during its curing process has been employed. We predicted the temperature behavior for separating two layers of nanocomposites materials. Mathematical modelling to represent the problem were discretized numerically by employing central Finite Different Method (FDM) with weight parameter θ. The Linear System Equation (LSE) resulted from the discretization are solved by three numerical methods, Alternating Group Explicit (AGE), Red Black Gauss Seidel (RBGS) and Jacobi (JB). Results acquired from RBGS and JB methods have become the benchmarks for evaluating the results from AGE methods. Then, we developed parallel algorithms on the mathematical model based on AGE, RBGS and JB methods. Finally, we evaluated and analyzed the performance indicators based on the parallel performance evaluation (PPE) in solving the PDE. Parallel algorithm is computed in Parallel Virtual Machine (PVM) programming on Linux operating system (Fedora 21) as a platform in order to accelerate the sequential execution as well as the convergence rate. Parallel AGE has significantly outperformed their counterparts compared with the benchmark classical methods (RBGS and JB) for two dimensional problems. Despite having huge computational complexity, results obtained from Parallel AGE methods have proven valuable since it enables to provide solution with higher accuracy and stability.

AGE Iterative Method with First-Order Quadrature Scheme for Fuzzy Fredholm Integral Equations of Second Kind

In this study, we propose Alternating Group Explicit (AGE) iterative method to solve the problem of linear fuzzy Fredholm integral equations of the second kind (FFIE-2) based on Trapezoidal quadrature rule. First, we generate a system of linear equations by using approximation equations. Next, we use AGE iterative method to solve the generated linear system of FFIE-2. Then, we illustrate the applicability of AGE iterative method on some numerical examples. Finally, we do the comparison to test the efficiency of AGE with Jacobi and Gauss-Seidel (GS) iterative method based on three parameters: number of iterations, execution time and Hausdorff distance. From our findings, we found that AGE is a better iterative method compared with Jacobi and GS to solve FFIE-2.

The Piecewise Polynomial Collocation Method for The Solution of Fredholm Equation of Second Kind By Using AGE Iteration

In this study, we study a piecewise approximate solution of Fredholm integral equation of the second kind by using the collocation discretization scheme. The discretization scheme has been used to build up a piecewise approximation equation which is the approximation equation form a system of linear equation. In the effort to get the numerical solution, the Gauss-Seidel iterative method and Alternating Group Explicit (AGE) have been treated as a linear solver. In addition to that, the formulation and the implementation of AGE method are included. Based on the numerical finding, the result showed that Alternating Group Explicit (AGE) iterative method is better than the Gauss-Seidel method.

Caputo’s Finite Difference Solution of Fractional Two-Point BVPs Using AGE Iteration

The aim of this paper deals with the performance of Alternating Group Explicit (AGE) iterative method for solving the linear system based on Caputo’s finite difference approximation equation. Actually, the approximation equation was be generated from the discretization process of the fractional two-point boundary value problems by using finite difference discretization scheme and Caputo’s operator. The formulation and implementation of AGE iterative method have been included. To investigate the efficiency of AGE iterative method, this paper considered four examples of numerical experiments. Based on numerical results, the findings showed that the AGE method is more efficient as compared with GS method.

Frequency-domain and time-domain solvers of parabolic equation for rotationally symmetric geometries

Both the frequency-domain and the time-domain body of revolution parabolic equations (FD-BoR-PE, TD-BoR-PE) are derived in this paper. By taking advantage of the rotationally symmetrical property, better performance of the PE method can be achieved for the analysis of bodies of revolution (BoRs). In each transverse plane, only the unknowns of a line, which starts from the center and passes through the node on the generatrix, need to be calculated. Then the unknowns for each transverse plane can be obtained by Fourier series summation for each Fourier mode and the calculation can be taken with a marching manner along the paraxial direction of the PE. As a result, the computational resources can be reduced greatly when compared with the traditional CN, alternating direction implicit (ADI) and alternating group explicit (AGE) finite difference schemes. Both the propagating and scattering problems are given to demonstrate the validity and efficiency of the proposed methods.

A Novel Parallel Parabolic Equation Method for Electromagnetic Scatterings

The alternating group explicit (AGE) scheme, together with the alternating direction implicit (ADI) scheme, is used for the parabolic equation (PE) method to analyze the electromagnetic scattering from electrically large conducting targets. The calculation can be taken plane by plane by using the finite difference scheme along the paraxial direction. Furthermore, each transverse plane is divided into two regions, which are a small region around the target and an outer region combined with the air layer and a perfect matching layer. The ADI scheme is applied to the small region, whereas the AGE scheme is introduced to the outer region. In this manner, only a few matrix equations are needed to be solved for the small region and the unknowns of the outer region can be directly obtained without solving any matrix equations. Therefore, the CPU time can be reduced significantly by the proposed method and it can be easily adapted to the parallel technology. The numerical results verify the accuracy and efficiency of the proposed method compared with the Crank–Nicolson-based PE method and the ADI-based PE method. Moreover, the parallel efficiency is given as the last numerical example.

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.

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.

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.

High Performance Large Sparse PDEs with Parabolic and Elliptic Types Using AGE Method on DPCS

The computational analysis of three case studies using parallelization of Alternating Group Explicit (AGE) solver is presented. Based on (2×2) block system and splitting strategy, AGE with Douglas-Richford and Brian variances are applied to simulate the large sparse PDEs applications with parabolic and elliptic types. The applications are heat equation, food dehydration for preservation and breast cancer growth. The AGE method has proved to be stable and suitable for parallel computing as it possesses separately and independently. The performance of AGE is compared with classical iterative methods such as Red Black Gauss Seidel (RBGS) and Jacobi (JB) methods. Since the PDEs applications are large sparse problems, we apply the AGE method in three different applications with three different mathematical models. The parallel implementation is based on SIMD model and supported by distributed memory architecture. Therefore, some numerical analysis and parallel performance indicators are used to validate the superior of parallel AGE method in terms of time execution, speedup, efficiency and effectiveness. As a result, the performances of numerical analysis and parallel evaluation of AGE are found to be effective for solving three case studies in reducing data storage accesses and minimizing communication time on a distributed parallel computer system.

Parallel Performance Comparison of Alternating Group Explicit Method Between Parallel Virtual Machine and Matlab Distributed Computing for Solving Large Sparse Partial Differential Equations

This paper presents the parallel implementation and performance analysis of a large sparse matrix of one dimensional parabolic Partial Differential Equation (PDE) between Parallel Virtual Machine (PVM) and Matlab Distributed Computing (MDC) platform. PVM is a software to perform parallel programming using heterogeneous or homogeneous clusters of parallel computing system connected by a logical network system to users as virtual memory parallel computer system. MDC is a scientific library programming language. Alternating Group Explicit (AGE) method is used to solve the parabolic equation with large sparse matrix. The distributed parallel algorithm of AGE method is implemented on the PVM with RedHat Linux operation and MDC Version 2011a. The parallel performance evaluation (PPE) is compared between the two platforms in terms of time execution, speedup, efficiency, effectiveness, and temporal performance. From the performance analysis, we can conclude that PVM is better compared to MDC.

Performance of HSAGE method with Seikkala derivative for 2-D fuzzy Poisson equation

In this study, iterative methods particularly families of Alternating Group Explicit (AGE) 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 AGE (FSAGE) and Half-Sweep AGE (HSAGE) methods were also presented. Then numerical experiments are carried out onto two example problems to verify the effectiveness of the methods.

The Arithmetic Mean Solver in Lagged Diffusivity Method for Nonlinear Diffusion Equations

Th is paper deals with the solution of nonlinear system arising fro m finite difference discretization of nonlinear diffusion convection equations by the lagged diffusivity functional iteration method co mbined with d ifferent inner iterative solvers. The analysis of the whole procedure with the splitt ing methods of the Arith met ic Mean (AM) and of the Alternating Group Exp licit (A GE) has been developed. A comparison in terms of number of iterations has been done with the BiCG-STA B algorith m. So me nu merical experiments have been carried out and they seem to show the effectiveness of the lagged diffusivity procedure with the Arithmet ic Mean method as inner solver.

High-accuracy cubic spline alternating group explicit methods for 1D quasi-linear parabolic equations†

In this article, we study the application of the alternating group explicit (AGE) and Newton-AGE iterative methods to a two-level implicit cubic spline formula of O(k 2+kh 2+h 4) for the solution of 1D quasi-linear parabolic equation u xx =φ (x, t, u, u x , u t ), 0<x<1, t>0 subject to appropriate initial and natural boundary conditions prescribed, where k>0 and h>0 are mesh sizes in t- and x-directions, respectively. The proposed cubic spline methods require 3-spatial grid points and are applicable to problems in both rectangular and polar coordinates. The convergence analysis at advanced time level is briefly discussed. The proposed methods are then compared with the corresponding successive over relaxation (SOR) and Newton-SOR iterative methods both in terms of accuracy and performance. This research work is dedicated to Late Prof. D.J. Evans

Alternating group explicit iterative method for nonlinear singular Fredholm Integro-differential boundary value problems

We discuss alternating group explicit (AGE) iterative method for the numerical solution of linear and non-linear singular integro-differential boundary value problems. The proposed AGE iterative method shows the superiority over the corresponding successive over relaxation (SOR) iterative methods. The presented algorithm is suitable for the use on parallel computers. The convergence analysis is briefly discussed. Computational results are presented to illustrate the utility of the proposed method.

The alternating group explicit (AGE) iterative method for solving a Ladyzhenskaya model for stationary incompressible viscous flow

In this paper, the alternating group explicit (AGE) iterative method is applied to a nonlinear fourth-order PDE describing the flow of an incompressible fluid. This equation is a Ladyzhenskaya equation. The AGE method is shown to be extremely powerful and flexible and affords its users many advantages. Computational results are obtained to demonstrate the applicability of the method on some problems with known solutions. This paper demonstrates that the AGE method can be implemented to approximate solutions efficiently to the Navier–Stokes equations and the Ladyzhenskaya equations. Problems with a known solution are considered to test the method and to compare the computed results with the exact values. Streamfunction contours and some plots are displayed showing the main features of the solution.