The Crank-Nicolson Method, CN

This method derives from the trapezium method in the ode field in which the time derivative in (8.9), expressed exactly as in (8.10), becomes a second-order central difference by virtue of the fact that the right-hand side now refers to a point in time midway in the time interval. This is achieved by taking the average of the second spatial derivative at the present time T and that at T - - 8T  

The result is a system exactly as (8.11) but with different definitions of the coefficients  

When applied to the solution of odes, the BI method (Chap. 4) uses a backward difference for the derivative on the left-hand side of (8.9) and the argument of the function on the right-hand side is the future, unknown, concentration vector. In our notation, at the point along the row of concentrations, this is 

This was formulated by Laasonen [343] in 1949. Rewriting this, the equation set becomes 

The solution of the above system of (8.11) will be described below, together with that for the CN method. 

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