Unequal Time Intervals

Just as space can be divided into unequally spaced intervals, so might time also be unevenly divided. As with spatial intervals, there is the choice between discretising on an uneven time grid or using a transformation to a new time scale. Since, except for BDF methods, one usually differentiates with respect to time using only two time points (levels), transformation does not make sense here. 

There is no problem with varying time intervals with two-level simulation methods, but with a method like BDF, there is the problem that one needs multipoint time derivatives calculated from unequally spaced points in time. Feldberg and Goldstein [36] show how to do this and even show how to apply the Feldbergian correction of half a time interval in this case, that becomes necessary when using the simple start for BDF, described in Chap. 4 (see also the consistency proof for this procedure in Appendix C). 

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Christensen, M. K., Determining the Parameters of First-Order Decay with a Nonzero End Point and Unequal Time Intervals, Anal. Chem. 55, 1983, 2324-2327. 

Another source of error is that errors in the beginning integrals tend to throw off all the predicted values of the dependent variables because the predicted values are obtained by summing the integrals up to the time of interest Thus, it would seem that the use of unequal time intervals with more data at short times is important in obtaining good precision. 

It is worth highlighting that, analogously to the case of non-uniform space grids, the use of unequal time intervals introduces a new dependence in the coefficients for the Thomas algorithm (a, j3 and 7 given by Eqs. (4.5)-(4.7) with AT = AT/ ) which are now different for each timestep since the time interval AT is different for each k value. Consequently, the Thomas... 

For unequal time intervals, we look at the example transformed diffusion equation 5.85. This is to be discretised at intervals of SO and H. We use index j to count 0 intervals and i to count X intervals (of H). 

Divide medium (0,1) into N equally spaced divisions (Figure 3-14). Let, = / A, with 0 = 0 and jv= 1 being the two boundaries. Let xy= Ax with equally spaced time interval. (In more advanced programming, one may also divide the time and space into unequal parts.) Three algorithms are discussed below. Other algorithms may be numerically unstable. 

There are different improvements that can be included in the different methods presented here. Thus, multi-point approximations to the derivatives [1], the use of unequally spaced spatial and time intervals [1, 8,9], and the use of more sophisticated methods of time integration [1], among others, have been developed. 

A practical route to an initial esiimate of the Lagrange multipliers X/ s is via the Burg algorithm (104,105). This generates a spectrum of maximum entropy provided the times tr are equally spaced, tr = rht. In many applications (see Sec. IV.B), there are specific reasons to preferring unequally spaced time intervals. One option is to use the... 

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