The normalised alternating group explicit (nage) method

Alternating Direction Implicit (ADI) methods have been in use since 1954 for the solution of both parabolic and elliptic partial differential equations. The convergence of these methods can be dramatically accelerated when good estimates of the eigenvalues of the operator are available, However, in the case of computation on parallel computers, the solution of tridiagonal systems imposes an unreasonable overhead. We discuss methods to lower the overhead imposed by the solution of the corresponding tridiagonal systems. The proposed method has the same convergence properties as a standard ADI method, but all of the solves run in approximately the same time as the "fast" direction. Hence, this acts like a "transpose-free" method while still maintaining the smoothing properties of ADI. Algorithms are derived and convergence theory is provided.

Within the last decade, attention has been devoted to the introduction of several fast computational methods for solving the linear difference equations which are derived from the finite difference discretisation of many standard partial differential equations of Mathematical Physics. In this paper, the authors develop and extend an exact factorisation technique previously applied to parabolic equations in one space dimension to the implicit difference equations which are derived from the application of alternating direction implicit methods when applied to elliptic and parabolic partial differential equations in 2 space dimensions under a variety of boundary conditions.

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.

The main purpose of this paper is to examine the application of complexity reduction approach based on half-sweep iteration concept with Alternating Group Explicit (AGE) method namely Half-Sweep AGE (HSAGE) method for solving system of linear equations generated from the discretization of two-point fuzzy boundary value problems (FBVPs). To form the linear system, the corresponding second order finite difference scheme has been used to derive the half-sweep finite difference approximation equation of the problems. The formulation and implementation of the proposed iterative method are discussed. Furthermore, numerical results of two test problems are also included in order to verify performance of the method compared to Full-Sweep Gauss-Seidel (FSGS) and Full-Sweep AGE (FSAGE) methods. The findings show that the proposed HSAGE method is superior to other tested methods in the sense of number of iterations, execution time and Hausdorff distance.

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.

In this paper, the extension of the generalized finite difference method to the solution of anisotropic elliptical and parabolic partial differential equations is given in the case of considering two space dimensions. The explicit finite difference formulae and the criterion of stability for the anisotropic parabolic equations are given. The stability condition has been expressed in function of the coefficients of the star equation for irregular clouds of nodes. Various anisotropic cases of simulation of cardiac tissue, including elliptic and parabolic equations, have been solved, and the results show the accuracy of the method.

FIDISOL is a program package for the solution of nonlinear systems of two-dimensional and three-dimensional elliptic and parabolic partial differential equations (PDEs) with nonlinear boundary conditions (BCs) on the boundaries of a rectangular domain. A finite difference method (FDM) with an arbitrary grid and arbitrary consistency order is used, these are either prescribed by the user or are self-adapted for a given relative tolerance. FIDISOL has been designed to be fully vectorizable on vector computers. In this paper we discuss several problems from the viewpoint of software development and user interface, for example, how to deliver the PDEs and BCs to FIDISOL and how to allow a flexible use by a suitable parameter list.

In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in- volving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numeri- cal results are presented to assess our approach.

In this paper we present a two parameter alternating group explicit (AGE) iterative method(TAGE) to solve the tridiagonal linear system Ay = k derived from the finite difference discretisation of a 2 point boundary value problem. The proof for the convergence of both the two parameter AGE and single parameter AGE [2] methods when the coefficient matrix is unsymmetric and real is presented. The case for the best acceleration parameters is also studied. The AGE iterative methods are used to solve four differential equations. The two parameter AGE method is explicit, accurate and flexible.

Practically speaking, fluid flow in round and noncircular nozzle is a very regular occurrence. The cold and hot water used in our homes is delivered to us via pipes. Water is delivered throughout the city via large pipe networks. Large pipelines carry natural gas and oil hundreds of kilometres from one place to another. During the operation of an engine, cooling water is carried by hoses to the radiator’s pipes, in which it is cooled as it travels. Experimentally, we detected the results of the model because there is no restriction on the application of experimental research to a certain sector or kind of concept. It may be used for a broad range of events and circumstances. Under parabolic velocity conditions, fluid (Water Pr 6.9) flows from the inlet position. The top and bottom walls of the rectangular nozzle are also moving at the same velocity as they are at the inlet position. Due to the movement of the walls, fluid is compressed in the particular region and also exhibits the same parabolic behavior. The solution of the coupled equations is determined by using the Finite Volume Method (FVM). When partial differential equations are expressed as algebraic equations, the FVM may be used to evaluate them. It can be used to evaluate elliptic, parabolic, and hyperbolic partial differential equations. Using FVM, it is necessary to know the values (and derivatives) of multiple variables at the cell faces, when the values (and derivatives) of these variables are only known at the cell centres. When determining these variables for convective terms, it is common to take the direction of the flow into consideration. The numerical results of the velocity and the pressure could be seen in the rectangular nozzle.

A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions

The stochastic counterpart of conservation laws with heterogeneous conductivity fields: Application to deterministic problems and uncertainty quantification

In response to the growing use of reaction diffusion problems in many fields, this monograph gives a systematic treatment of a class of nonlinear parabolic and elliptic differential equations and their applications these problems. It is an important reference for mathematicians and engineers, as well as a practical text for graduate students.

This paper presents a meshfree collocation method that uses deep learning to determine the basis functions as well as their corresponding weights. This method is shown to be able to approximate elliptic, parabolic, and hyperbolic partial differential equations for both forced and unforced systems, as well as linear and nonlinear partial differential equations. By training a homogeneous network and particular network separately, new forcing functions are able to be approximated quickly without the burden of retraining the full network. The network is demonstrated on several numerical examples including a nonlinear elasticity problem. In addition to providing meshfree approximations to strong form partial differential equations directly, this technique could also provide a foundation for deep learning methods to be used as preconditioners to traditional methods, where the deep learning method will get close to a solution and traditional solvers can finish the solution.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.