Point-Iterative Methods

Consider a system of n equations and n unknowns in matrix form, 

This problem can be expressed in an element form where the coefficients of matrix A can be seen explicitly  

In the Jacobi method, we rearrange the system of equations to place the contribution due to Xi on the LHS of the ith equation and the other terms on the RHS, and we divide both sides of the equation by an. 

The iteration equation for the Gauss-Seidel method is obtained employing the last available values within the iteration process  

The convergence rate of the Jacobi and Gauss-Seidel methods depends on the properties of the iteration matrix. By experience, it has been found that these methods can be improved by the introduction of a relaxation parameter a. Consider the Gauss-Seidel method, it can be rewritten as  

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There are two basic families of solution techniques for linear algebraic equations Direct- and iterative methods. A well known example of direct methods is Gaussian elimination. The simultaneous storage of all coefficients of the set of equations in core memory is required. Iterative methods are based on the repeated application of a relatively simple algorithm leading to eventual convergence after a number of repetitions (iterations). Well known examples are the Jacobi and Gauss-Seidel point-by-point iteration methods. 

Unfortunately, the optimum value of the relaxation parameter is problem and grid dependent, hence it is difficult to give precise guidance. Besides, the convergence rate of the point-iterative methods rapidly reduces as the grid is refined. 

Multigrid acceleration of the Gauss-Seidel point-iterative method is currently used in many commercial CFD codes to solve the system of algebraic equations resulting from the discretization of the governing equations. For this reason, the basic principles and nomenclature must be known by the users of commercial codes and in particular for researchers that are making their own codes. 

Calculation of pH titn. curves and end-points. Iterative method with interval halving... 

This chapter has discussed methods for solving for the roots of a single equation. The emphasis has been on two methods, the successive substitution (or zero point iterative) method and Newton s method with the major emphasis on Newton s method. The approach has been different from many textbook treatments of this subject in that the emphasis has been on using Newton s method with a numerical... 

The general case has been solved by Bashforth and Adams [14], using an iterative method, and extended by Sugden [15], Lane [16], and Paddy [17]. See also Refs. 11 and 12. In the case of a figure of revolution, the two radii of curvature must be equal at the apex (i.e., at the bottom of the meniscus in the case of capillary rise). If this radius of curvature is denoted by b, and the elevation of a general point on the surface is denoted by z, where z = y - h, then Eq. II-7 can be written... 

The method is applicable at reduced temperatures above 0.30 or the freezing point, whichever is higher, and below the critical point. The method is most reliable when 0.5 prediction average 3.5 percent when experimental critical properties are known. Errors are higher for predic ted criticals. The method is useful when solved iteratively with Eq. (2-23) to predict the acentric factor. 

Elliptic Equations Elhptic equations can be solved with both finite difference and finite element methods. One-dimensional elhptic problems are two-point boundary value problems. Two- and three-dimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider howthe equations are discretized to form sets of algebraic equations and howthe algebraic equations are then solved. 

Some situations, however result in the form of second-order diffferential equations, which often give rise to problems of the split boundary type. In order to solve this type of problem, an iterative method of solution is required, in which an unknown condition at the starting point is guessed, the differential equation integrated twice and the resulting solution compared with a known boundary condition, obtained at the end point of the calculation. Any error between the known value and the calculated value can then be used to revise the initial starting guess for the next iteration. This procedure is then repeated until... 

Because this method avoids iterative calculations to attain the SCF condition, the extended Lagrangian method is a more efficient way of calculating the dipoles at every time step. However, polarizable point dipole methods are still more computationally intensive than nonpolarizable simulations. Evaluating the dipole-dipole interactions in Eqs. (9-7) and (9-20) is several times more expensive than evaluating the Coulombic interactions between point charges in Eq. (9-1). In addition, the requirement for a shorter integration timestep as compared to an additive model increases the computational cost. 

On the subject of comparing iterative methods a word of caution is in order. Clearly in any quantitative comparison, the termination criteria should be comparable and the benchmark problems should be run on the same computer. Yet even for simple problems and methods, these two requirements prove to be difficult to enforce and insufficient to ensure meaningful comparisons. To allow for the fact that different methods do not terminate at exactly the same point even when the same termination criterion is used, Broyden (B13) introduced a mean convergence rate, R, which is... 

Finally, a special point to look for in comparing iterative methods for pipeline network problems is to use the same problem formulation for both methods otherwise the results may reflect differences in formulations as well as iterative methods. 

The examples in the previous section demonstrate that nonunique solutions to the equilibrium problem can occur when the modeler constrains the calculation by assuming equilibrium between the fluid and a mineral or gas phase. In each example, the nonuniqueness arises from the nature of the multicomponent equilibrium problem and the variety of species distributions that can exist in an aqueous fluid. When more than one root exists, the iteration method and its starting point control which root the software locates. 

Well-known iterative methods for model calibration of the OUR curve can be used to determine these two constants when results from procedures number 1 and 2 are available. The model used for the determination of these parameters is based on the matrix formulated in Table 7.1. The values shown for khn and KXn in Table 6.7 may be used as a starting point for this iteration. 

These nonlinear equations must be solved simultaneously at each point in time. Usually an iterative method is used and sometimes convergence problems occur. The complexity grows as the number of chemical species increases. 

Generally, the weak points of methods should be quickly identified and eliminated, thus every validation parameters should be tested as early as possible. If the method fails with respect to one of the parameters, the entire method has to be changed. Consequently, the validation must be started again. The development and validation may be an iterative process (Diagram 1). However, it is beneficial to keep the iterative steps to a minimum. 

This is called point-simultaneous overrelaxation. If we set k = [s]nn, we have obtained the discrete formulation of Van Cittert s method. This connection between Van Cittert s method and the classic iterative methods of solving simultaneous equations was demonstrated in an earlier work (Jansson, 1968, 1970). 

The nonlinear iterative methods described here are based on the linear relaxation methods developed in Sections III. C.l and III.C.2 of Chapter 3. Initially, the correction term was set equal to zero in regions where o(k) was nonphysical. To illustrate this, we may rewrite the point-simultaneous equation [Chapter 3, Eq. (23)] with a relaxation parameter that depends on the estimate d(k) ... 

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