Crank-Nicolson CN scheme

We assume, for a start, the simple diffusion equation 5.12. We have seen in Sect. 5.1, that the normal explicit method, with its forward-difference discretisation of 8c/3t performs rather poorly, with an error of 0(6t). The discrete expression for the second derivative (right-hand side of Eq. 5.12) is better, with its error of 0(h ). Let us now imagine a time t+ig6t at this time, the discretisation 

Note that we are not explicitly interested in the values at t+ St but Eq. 5.28 is a statement using that time as a centre of symmetry with good approximations on both sides. Eq. 5.28 rearranges, for all i to the form 

In Chapt. 6 we show that simple CN can perform badly in certain cases which, unfortunately, comprise the majority of all experiments that need simulating. So why go on with this method Fortunately, the problem can be overcome, with a little extra effort (Sect. 5.2.2). 

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This system of coupled equations must be solved by integrating forward in time starting from the initial conditions mk,i(0). Eor stability, the time integration of the diffusion term can be treated implicitly using, for example, the Crank-Nicolson (CN) scheme. If we denote the volume-average moments at time t = n At by , a semi-implicit scheme for... 

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