Alternating direction implicit ADI method

Alternating direction implicit (ADI) method 161 These methods are two-step methods involving the solution of traditional systems of equations along lines parallel to the x- and y-axes, at the first and 

In the Peaceman-Rachford ADI method, in the first step, it is advanced from to t +1/2, and in the second step, it is advanced from 1/2 to t +1-The ADI method is written as 

The boundary conditions to obtain the solution of Equation 1.135a can be obtained as 

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Eliminating from Equation 1.135a and Equation 1.135b, 

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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. 

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. 

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

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... 

Peaceman and Rachford " developed an Alternate Direction Implicit (ADI) method, which retains a tridiagonal matrix and allows two-dimensional equations... 

Equation (140) can be solved either using an iteration method such as the Gauss-Siedel method or a direct method such as tridiagonal-matrix algorithm (TDMA). Other methods, such as the Alternating-Direction Implicit (ADI) method combine iteration with the TDMA. 

The differential equation for head pressure (2) is solved by over-relaxation iterative method [4] filtration rate is calculated from Darcy law by using defined values of hydraulic head. Transport equations of reagent concentration in liquid phase (5), useful element concentration in solid phase (4), and its transition to liquid phase (6) are solved together by the implicit Crank-Nicolson scheme. Crank-Nicolson scheme is implemented in three stages in case of 3D problem by using splitting technique of the alternating direction implicit (ADI) method [4]. 

Alternating-direction-implicit (ADI) methods. Let us eonsider the classical, dimensionless, two-dimensional heat equation for u(x,y,t),... 

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