Laasonen method

Implicit methods have the great advantage of being stable, in contrast with the explicit method. It will be seen (and analysed in detail in Chap. 14) that the Laasonen method, a kind of BI, is very stable and responds to sharp transients with smoothly declining (but relatively large) errors, whereas Crank-Nicolson, also nominally stable, responds with error oscillations of declining amplitude, but is highly accurate. The drawbacks of both methods can be overcome, as will be described below. 

The Laasonen method, because of the forward difference in T, has errors of 0(6T, H2), and the first-order behaviour with respect to ST limits its accuracy to about the same as the explicit method described in Chap. 5. However, it has a smooth error response to disturbances such as an initial transient (Cottrell), and is stable for any value of 6T/H2, where // is either the same as all intervals if equal intervals are used in X, or is the smallest (usually the first) intervai if unequal intervals are used. This makes the method interesting, and it will be seen below that it can be improved. For simplicity, the symbol A will be used below, and denotes the largest value of that parameter, that is, the value from the smallest interval in space in a given system. 

If a given her is of higher than first order, nonlinear terms arise in the dynamic equation(s). With terms, for example, in squared concentrations (see below), there is the danger, due to computational errors, that a concentration becomes negative, after which it can never be corrected. The technique CN is especially prone to this, because of the oscillations it engenders as a response to sharp transients such as a potential jump. This is one reason some workers prefer the Laasonen method or its improved offshoots, which have a smooth error response without any oscillations. With a Pearson start, however, CN can be used safely, without the appearance of negative concentrations. 

For the Laasonen method, C2 is expressed as the square of the next, unknown concentration, C 2. We have... 

This is an example of a Cottrell simulation using second-order extrapolation based on the BI (Laasonen) method and unequal intervals. Three-point spatial discretisation is used here. 

This program is again a Cottrell simulation using second-order extrapolation based on the Bl (Laasonen) method and unequal intervals, but in contrast with the above program C0TT EXTRAP, this one makes use of the four-point spatial derivative approximation, and the GU-function. It performs a little better than the above program, at little extra programming effort. 

Different meaning of chord AB in approximating d a/ dr in classic, Crank-Nicolson or Laasonen method (lower abscissa scale). 

This has been mentioned in several papers, but the first to describe such approximations were Mastragostino et al. in 1968 [57]. Let the squared term be C for example, a term in —KC in the dynamic equation. The change SC is equal to C —C. For the Laasonen method, is expressed as the square of the next, unknown... 

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