The Commonly Used Implicit Methods

Essentially, only two implicit methods will be described here, but with extensions that make them more useful. They are derived from the implicit methods described for odes in Chap. 4, BI and the trapezium method. These have different names in the pde context, as will be seen. 

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 discretisation of the second, spatial derivative of concentration will be reiterated in a general form that can then be built into the methods to follow. For the three concentrations grouped around the one at the point Xi, we can write the general linear expression, 

Dieter Britz Digital Simulation in Electrochemistry, Lect. Notes Phys. 000, 119—143 (2005) www.springerlink.com . Springer-Verlag Berlin Heidelberg 2005 

These a s are dependent on the index i but for brevity, this will not be indicated in what follows below. 

Strutwolf, Digital Simulation in Electrochemistry, Monographs 

If transformation is to be used, by the function (7.3) on page 125, then the resulting right-hand-side of the diffusion equation (7.5) can be written as a transformation of the second spatial derivative. 

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