Using KW as a Start for BDF

This works rather well with odes, as also seen in Fig. 4.7, where the 4-point BDF form was used. For use with pdes, however, it may be considered too much trouble to program, especially as there are easier options, for example, extrapolation, which produce results that are just as good. Also, if BDF is nonetheless chosen, it was found in Sect. 4.8.1 and proved mathematically in Appendix C that the simple start with a simple time correction produces rather good results for much less effort. 

Error orders are given in brackets (measured by taking 100 steps) 

Nevertheless, we compared the four starting methods simple, simple with 8t/2 correction ( simp+ ), rational and KW for the same simulation, and Table 4.1 shows the errors at f = 1 for each case. It is seen that for 3-point BDF, all but the simple start result in much the same error, but for higher BDF forms, the KW start outstrips the others impressively. 

The 3-point BDF KW start is actually very simple to implement, requiring only a 2 X 2 system whose solution (for yi and j2) is easily expressed, and so it could be feasible for use in pde. However, the table shows that it results in no better errors than simpH- or the rational start, so it does not recommend itself. It is interesting to note, regarding the error orders, that both simpH- and rational show an order close to 2, regardless of the BDF order, meaning that with these starts, BDF using more than three points is no improvement over three-point BDF. The only start that enables the full accuracy of higher BDF orders is the KW start, which follows the BDF order. 

All the techniques described above can also be applied to the numerical solution of systems of odes, and here we are getting closer to what happens when we solve pde%, because in effect, one reduces them to ode systems when discretising them. 

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