Stiff stability of BDF methods

we have studied in detail the stability properties of simple single-step methods of the class (4.150). More generally, explicit methods always have time restrictions of the form (4.169), and the more a method makes use of past data to predict the fiiture, the more strict the time step restrictions become. We consider here the stabiUty properties of implicit multi-step BDF methods 

The coefficients are compnted from (4.138) by substimting the Hermite interpolation polynomial (4.134) into the update formula (4.130). An efficient implementation of BDF methods is presented below in the discnssion of DAE systems. For the test equation jc = Ax, Ah = we again expand the states in the numerical trajectory in eigenvectors of A, and define the growth coefficient for each mode as 

Equating the sum in the braces to zero, defining again coj = -(At)Xj, and multiplying by 

From our discussion of the discretized time-dependent diffusion equation (4.174), we have seen that the fastest modes have spatial wavelengths comparable to the distance between 

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