Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 38C / b

The Dahlquist equivalence theorem states that a linear multistep method is convergent for well-posed sufficiently smooth initial-value problems, with consistent starting values, if and only if it is consistent and zero-stable. Zero-stability means that every root of its first characteristic polynomial lies in the closed unit disk, with roots on the unit circle simple.

Here the characteristic polynomials areConsistency holds because $\rho (1)=0$ and ${\rho }^{\prime }(1)=\sigma (1)=2a$. The two remaining roots have product one. For $0<a<2$ they are distinct conjugate unit roots, and neither equals one. At $a=0$, the root one is triple; at $a=2$, the root minus one is double. Outside $[0,2]$, the remaining roots are real reciprocal roots and one has modulus greater than one. ThusThe order conditions give $c_2=0$ and $c_3=6-a$, which is nonzero throughout the convergent range. Therefore the order of convergence is exactly two, for starting approximations of matching accuracy.