Numerical solution alternating direction implicit

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 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. 

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... 

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... 

Jim Douglas, Jr., A note on the alternating direction implicit method for the numerical solution of heat flow problems, Proc. Amer. Math. Soc. vol. 8 (1957) pp. 409-412. 

In discussing the alternative theoretical approaches let us limit ourselves to those which have been applied directly to processes in which we are interested in this article, but first of all let us stress once more the importance of the work of Delos and Thorson (1972). They formulated a unified treatment of the two-state atomic potential curve crossing problem, reducing the two second-order coupled equations to a set of three first-order equations. Their formalism is valid in the diabatic as well as the adiabatic representation and also at distances of closest approach near Rc. Moreover the problem of the residual phase x(l) is solved implicitly. They were able to show that a solution of the three first-order classical trajectory equations is not sensitive to all details of the potentials and the coupling term, but to only one function which therefore can be used readily for modelling assumptions. The resulting equations should be solved numerically. Their method has been applied now to the problem of the elastic scattering of He+ + Ne (Bobbio et at., 1973) but unfortunately not yet to any ionization problem. 

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