Alternating direction implicit finite-difference

The numerical simulation for this study is based on the alternating direction implicit finite difference method, using an expanding... 

Alternating direction implicit finite difference method in conjunction with the Thomas algorithm has been applied. 

G. Sun and C. W. Trueman, Some fundamental characteristics of the one-dimensional alternating-direction-implicit finite-difference-time-domain method, IEEE Trans. Microw. Theory Tech., vol. 52, pp. 46-52, Jan. 2004.doi 10.1109/TMTT.2003.821230... 

Figure 3.19 Finite difference molecule for the alternating-directions implicit method (ADI) of solving the two-dimensional diffusion equation.

In our numerical model, Eq.(2.8) was transformed into a six-point finite-difference equation using the alternative direction implicit method (ADIM). At the edges of the computational grid (—X,X) radiation conditions were applied in combination with complex scaling over a region x >X2, where —X X j) denotes the transverse computational window. For numerical solution of the obtained tridiagonal system of linear equations, the sweep method" was used. 

The contaminant transport model, Eq. (28), was solved using the backwards in time alternating direction implicit (ADI) finite difference scheme subject to a zero dispersive flux boundary condition applied to all outer boundaries of the numerical domain with the exception of the NAPL-water interface where concentrations were kept constant at the 1,1,2-TCA solubility limit Cs. The ground-water model, Eq. (31), was solved using an implicit finite difference scheme subject to constant head boundaries on the left and right of the numerical domain, and no-flux boundary conditions for the top and bottom boundaries, corresponding to the confining layer and impermeable bedrock, respectively, as... 

The numerical solution to the advection-dispersion equation and associated adsorption equations can be performed using finite difference schemes, either in their implicit and/or explicit form. In the one-dimensional MRTM model (Selim et al., 1990), the Crank-Nicholson algorithm was applied to solve the governing equations of the chemical transport and retention in soils. The web-based simulation system for the one-dimensional MRTM model is detailed in Zeng et al. (2002). The alternating direction-implicit (ADI) method is used here to solve the three-dimensional models. 

Menon and Landau [52] developed a model to describe transient diffusion and migration in stagnant binary electrolytes. Nonuniformity at a partially masked cathode was found to increase during electrolysis as the diffusion resistance develops. The calculations were done using an alternating-direction implicit (ADI) finite difference method. 

It was solved numerically using the alternating-direction implicit (ADI) finite difference method (5). The steady-state results were obtained as a long time limit and presented in the form of two-parameter families of working curves (5). These represent steady-state tip current or collection efficiency as functions of K = akc/D and L. 

Another approach involves using implicit methods (28, 30, 31) for obtaining/(y, k + 1) [e.g., the Crank-Nicolson (32), the //y implicit finite difference (FIFD) (33), and the alternating-direction implicit (ADI) (34) methods] rather than the explicit solution in (B.1.9). In implicit methods, the equations for calculation of new concentrations depend upon knowledge of the new (rather than the old) concentrations. There are a number of examples of the use of such implicit methods in electrochemical problems, such as in cyclic voltammetry (35) and SECM (36). 

The alternating direction implicit method (i.e., ADI) is employed to calculate transverse diffusion in the x direction via second-order-correct finite differences for a second derivative using unknown molar densities at Zk+i-Hence,... 

Studies on solidification modeling have been largely directed towards macroscopic phenomena. A variety of numerical techniques have been used for such modeling studies. Among these are the finite difference method (FDM) with or without the alternate direction implicit (ADI) time-stepping scheme, the FEM, the boundary element method (BEM), the direct finite difference method (DFDM), and the control volume element (VFM) method. 

This problem was solved semianalytically in terms of two-dimensional (2D) integral equations [2,3] and numerically by using Krylov integrator [4] and the alternating direction implicit (ADI) finite difference method [3,5]. Potentiostatic transients were computed for two limiting cases a diffusion-controlled process and totally irreversible kinetics [3-5]. The analysis of the simulation... 

An alternative approach to the finite element approach is one, introduced as a concept by Courant as early as 1943 [197], in which the total energy functional, implicit in the finite element method, is directly minimized with respect to all nodal positions. The approach is conjugate to the finite element method and merely differs in its procedural approach. It parallels, however, methods often used in atomistic modeling schemes where the potential energy functional of a system (e. g., given by the force field ) is minimized with respect to the position of all (or at least many) atoms of the system. A simple example of this emerging technique is given below. 

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