The implicit difference method from J. Crank and P. Nicolson

3 The implicit difference method from J. Crank and P. Nicolson 

A particularly accurate implicit difference method, which is always stable, has been presented by J. Crank and P. Nicolson [2.65]. In this method the temperatures at the time levels tk and tk+l are used. However the differential equation (2.236) is discretised for the time lying between these two levels tk + At/2. This makes it possible to approximate the derivative (dt /dt)k+1 2 by means of the accurate central difference quotient 

This is advantageous because the choice of an implicit difference method allows larger time steps to be used and therefore requires a more exact approximation of the derivative with respect of time. 

The second derivative ( 92 d/ 9x2)f+1 2 at time tk + At/2 is replaced by the arithmetic mean of the second central difference quotients at times tk and tk+l. This produces 

The temperatures at time tk on the right hand side of (2.268) are known the three unknown temperatures at time tk+1 on the left hand side have to be calculated. The difference equation (2.268) yields a system of linear equations with i = 1,2. n. The main diagonal of the coefficient matrix contains the elements (2 + 2M) the sub- and superdiagonals are made up of the elements (—M) all other coefficients are zero. In this tridiagonal system, the first equation (i = 1) cannot contain the term —Mi q+1 and likewise, in the last equation (i = n) 

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