Kimble White, KW

The method due to Kimble Sz White [338] is not actually a method designed for odes, but was devised by the authors for electrochemical pdes. The method can however be easily adapted to odes and in fact might be more appropriate there. The method described in 1990 had a precursor in 1987 [414] and this section will start with a description of its expression for odes, because it is simpler and makes the point more clearly. A cut-down application of it has already been outlined in Sect. 4.8.1. 

The essence of KW is that multi-point central differences are used as derivatives along most of the t scale, with some asymmetric expressions necessarily added at the ends. Rather than using the time-marching method that is common to all the methods described in previous sections, KW puts all the approximations into one large system of equations, and solves the lot. It turns out that this results in a fortuitous stability [141]. 

The method is based on another time-marching scheme not mentioned in the above sections the leapfrog method, using central differences. Equation (4.1) can be approximated as 

The interesting thing is that all but the last equation are leap-frog forms, which by themselves result in an unstable solution the mere addition of the last (4.39) renders the system stable, and the solution is of 0(8t2). 

For i = 1, a 5-point asymmetrical form, called y2(5) in the Table, is applied to the point yi  

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