Damping the CN Oscillations

There are a few ways of subdividing the first time step using a number M of equal intervals, or a number of intervals expanding with time, usually exponentially. The two methods can be formally combined into one description. Let the full interval to be subdivided be of length ST, and let it be subdivided into M smaller intervals Ti,i = 1. ..M, such that 

Exponentially expanding subintervals are obtained if y 1 and equal intervals for y = 1. The latter method is (here) called Pearson, after the author who first suggested it [14], in 1965. It has been studied more recently [13, 15-18]. Exponentially expanding subintervals will be called ees here. They were suggested [19] and later used [20-22], and studied in some detail recently [15,18]. 

Whether Pearson (in the one form or the other) or ees is to be used is a matter of taste. Pearson is the simpler method, and it is simpler to determine the only parameter involved, M. Numerical experiments [15] show that a (sub)A value of about unity is sufficient to damp oscillations during the first M substeps, so this sets M simply to such a value as to satisfy the requirement. That is, if the main time interval ST leads to A 1, then A subintervals are needed in order to bring the sub-A below unify, or Af = A, but rounded up to the nearest integer. Besides simplicity, this has the additional advantage of equal time intervals in many simulations, the coefficients as in (8.14) depend on the time interval, and must be recalculated 

For ees, there is no simple recipe for the choice of M and y. The reader is referred to the study [15], where several contour plots are provided that can help. A rough guide is that y = 1.5 is a fairly universally useful value. It is the opinion of the present authors that Pearson is the best choice here. 

If ees is considered desirable, there is the small matter of the determination of the parameters. Normally, one would choose Af first, and then either the size of the first interval ti (which sets the expansion parameter y) or y, which sets the first interval. In the former case, having chosen M and ti, the function EE FAC (see Appendix E) can then be used to find the appropriate y. In the latter case, Eq. (7.19) on page 129 can be inverted to give explicitly 

Crank and Nicolson, in their original paper [185], recognised the oscillation problem with their method, writing If 7 [which is their 6t/6x2 J is very large an oscillatory error which only disappears very slowly may arise . The problem is referred to in most texts describing the method. More detail is given in Chap. 14, but the essence of the problem is that CN will oscillate if A 0.5  

Whether Pearson (in the one form or the other) or ees is to be used, is a matter of taste. Pearson is the simpler method, and it is simpler to determine the only parameter involved, AI. Numerical experiments [149] show that a 

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