Difference approximation, Crank-Nicholson

When faster reactions are dealt with, it may be profitable to remove the At/Ay2 < 0.5 condition and use an implicit method such as the Crank-Nicholson method.15 17 The finite difference approximation is then applied at the value of t corresponding to the middle of the j to j + 1 interval, leading to... 

Ehrlich (El) uses the Crank-Nicholson (C16) finite-difference procedure for the integration of the diffusion equation, with a three-point approximation of the space derivatives on either side of the moving... 

Denoting the solute species in a binary system by subscripts 1 and 2, the Crank-Nicholson finite difference approximation for... 

Traditional finite difference methods [55, 81] for solving time-dependent second-degree partial differential equations (such as modified diffusion equation) include forward time-centered space (ETCS), Crank-Nicholson, and so on. For time-independent second-degree partial differential equations such as Poisson-Boltzmann equation, finite difference equations can be written after discretizing the space and approximating derivatives by their finite difference approximations. For space-independent dielectric constant, that is, E(r) = e, a tridiagonal matrix inversion needs to be carried out in order to obtain a solution for tp for a given/. 

This again leads to a set of coupled equations that are tri-diagonal in nature. Since this method used a centered time difference approximation, it is accurate to second order in both time and space while the FD and BD methods are only accurate to first order in the time variable. Thus one would expect an improved accuracy with the Crank-Nicholson method. However, numerical stability of the three techniques is perhaps more important than accuracy. 

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