Implicit alternating direction

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. 

Since the energy balance involves second-order derivatives in both the x and y directions, the alternating direction implicit (ADI) method is used (92). This method requires three time levels of temperature and involves the solution of the equation twice, once in each direction. 

Another type of nondisk-shaped SECM tips are UMEs shaped as spherical caps. They can be obtained, for example, by reducing mercuric ions on an inlaid Pt disk electrode or simply by dipping a Pt UME into mercury [15]. An approximate procedure developed for conical geometry was also used to model spherical cap tips [12]. Selzer and Mandler performed accurate simulations of hemispherical tips using the alternative direction implicit final difference method to obtain steady-state approach curves and current transients [14]. As with conical electrodes, the feedback magnitude deceases with increasing height of the spherical cap, and it is much lower for a hemispherical tip than for the one shaped as a disk. 

Avery powerful method for the solution of system 8-68, widely in use, is the so-called alternating-direction implicit method, which is based on the idea of splitting each timestep... 

The alternating direction implicit methods, or ADI methods, is a method of variable direction. This method employs a splitting of the time step to obtain a multi-dimensional implicit method which requires only the inversion of a tridiagonal matrix. 

The alternating direction implicit method of finite differencing Is a method of variable direction in finite differencing Employs splitting of one time step into two to obtain implicit method Requires only inversion of tridiagonal matrix in this method. 

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. 

The alternating direction implicit (ADI) method (Peaceman and Rachford, 1955) is a partially implicit method. The equation is rearranged so that one coordinate may be solved implicitly using the Thomas algorithm whilst the others are treated explicitly. If this is done alternately, each coordinate has a share of the implicit iterations and the efficiency (Gavaghan and Rollett, 1990) as well as the stability is improved. The method was used by Heinze for microdisc simulations (Heinze, 1981 Heinze and Storzbach, 1986) and has subsequently been adopted by others (Taylor et al, 1990 Fisher et al., 1997). 

A totally different approach to solving the Navier-Stokes equations is made in alternating direction implicit (ADI) methods (Briley and McDonald, 1975) and approximate factorization implicit (AFI) methods (Beam and Warming, 1977, 1978). These methods apply an approximate spatial factorization technique to avoid the inversion of huge banded matrices and are computationally very efficient. 

ADI alternating-direction implicit CAD computer-aided design... 

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. 

The wave function is propagated forward in time via a Peaceman-Rachford alternating-direction, implicit scheme [6] given by... 

This equation is to be formulated for all grid points (i, j). A system of linear equations for the unknown temperatures at time tk+l, that has to be solved for every time step, is obtained. Each equation contains five unknowns, only the temperature at the previous time tk is known. A good solution method has been presented by P.W. Peaceman and H.H. Rachford [2.69]. It is known as the alternating-direction implicit procedure (ADIP). Here, instead of the equation system (2.305) two tridiagonal systems are solved, through which the computation time is reduced, see also [2.53]. 

Several strategies have been proposed to treat these interfacial coupling terms in the design of appropriate numerical solution methods. A simple way to implement these interphase coupling terms is to apply an explicit discretization scheme. Alternatively, an alternating direction implicit (ADI)-like method can be applied in which the current phase variable in the coupling term can... 

ADI Alternating Direction Implicit ADM Axial Dispersion Model AS MM Algebraic Slip Mixture Model... 

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. 

The alternating direction implicit (ADI) method combined with a modified Thomas algorithm was used. 

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. 

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... 

The coupled set of nonlinear differential equations (equations 1 and 4) are solved by the alternating direction implicit (ADI) method f9-10) on an evenly spaced grid. The advective transport of a solute species was solved using the Lax-Wendroff two-step method (10). To ensure that numerical dispersion is avoided, a grid spacing was chosen such that the grid Peclet number (defined by < 2 fll). The computational expense involved in using a... 

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). 

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