Implicit finite-difference algorithm

Beam, R. M., and Wanning, R. R, An implicit finite difference algorithm for hyperbolic systems in conservation-law form. J. Comp. Phys. 22(1), 87 (1977). 

Rudolph, M. (1991) A fast implicit finite difference algorithm for the digital simulation of electrochemical processes, JJElectroanal.Chem. 314, 13-22. 

Rudolph M (1995) Digital simulations with the fast implicit finite difference algorithm - the development of a general simulator for electrochemical processes. In Rubinstein I (ed) Physical electrochemistry. Principles, methods, and applications. Marcel Dekker, New York, pp 81-129... 

Shoup D, Szabo A (1986) Explicit hopscotch and implicit finite-difference algorithms for the Cottrell problem exact analytical results. 

Mocak J, Feldberg SW (1994) The Richtmyer modification of the fully implicit finite difference algorithm for simulations of electrochemical problems. J Electroanal Chem 378 31-37... 

Feldberg SW, Goldstein Cl (1995) Examination of the behavior of the fully implicit finite-difference algorithm with the Richtmyer modification behavior with an exponentially expanding time grid. J Electroanal Chem 397 1-10... 

The finite difference algorithm is obtained by replacing the time derivatives by a forward difference, using an implicit rule to evaluate F(Y) at time t , and setting h = At. The result is... 

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

Bieniasz LK, 0steiby O, Blitz D (1995) Numerical stability of finite difference algorithms for electrochemical kinetic simulations matrix stability analysis of the classic explicit, fully implicit and Crank-Nicolson methods and typical problems involving mixed boundtiry conditions. Comput Chem 19 121-136... 

To obtain an algorithm that is unconditionally stable, we consider an implicit discretization scheme that results from using backward finite-differences for the time derivative. The corresponding difference equation is most conveniently obtained by approximating the diffusion equation at point (Xj,tn+i) ... 

Methods applying reverse differences in time are called implicit. Generally these implicit methods, as e.g. the Crank-Nicholson method, show high numerical stability. On the other side, there are explicit methods, and the methods of iterative solution algorithms. Besides the strong attenuation (numeric dispersion) there is another problem with the finite differences method, and that is the oscillation. 

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. 

There are a number of numerical algorithms to solve the difference equation representation of the partial differential equations. Implicit algorithms such as Crank-Nicolson scheme where the finite difference representations for the spatial derivatives are averaged over two successive times, t = nAt and t = n + l)At, are frequently used because they are usually unconditionally stable algorithms. Most conservation laws lead to equations of the form... 

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