Trapezium method

This method seems at first sight unpromising, because of its low error order, the same as that for Euler. However, it has some very useful stability properties (see later) and forms the basis for several high-order methods, as will be seen. 

We know from (3.13) in Chap. 3, how that same derivative approximation is of higher order 0(8f) when applied at the midpoint, and this leads to the trapezium method, in which we must find an expression for the right-hand side of (4.1) at time t + j8t. This can be approximated as the average of the values at both ends  

This can be awkward to go on with, being implicit in y +, in our specific example (4.3), however, there is no problem, the above equation becoming 

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It will be seen that for the three methods Euler, BI and the trapezium method, the same approximation expression is used for the left-hand side of (4.1) but because of points made in questions (2) and (3) above, the methods are very different. 

In principle, all the methods described above for single odes can be used for the solution of such a system, when extended suitably. In the case of explicit methods such as Euler or RK, this is very simple to implement, whereas with implicit methods such as BI or the trapezium method, there are some choices to be made. 

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. 

This method derives from the trapezium method in the ode field in which the time derivative in (8.9), expressed exactly as in (8.10), becomes a second-order central difference by virtue of the fact that the right-haud side now... 

In the case of the ultramicroelectrodes such as the disk electrode, it is necessary to integrate over the surface, and sometimes there will be unequally spaced points along the surface, as for example, in direct discretisation on an unequal grid in the example program UME DIRECT. As mentioned in Chap. 12, it is found that due to the errors in the computed concentration values, the local fluxes are so inaccurate that any integration method better than the simple trapezium method is not justified. The routine U TRAP is thus recommended here. It integrates local current densities, precalculated by using the above routine U DERIV. 

The last way to be mentioned here to discretize the ode (21) is what is called the trapezium method ... 

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