Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 17C / b / Solution

The Dahlquist equivalence theorem says that a consistent linear multistep method is convergent exactly when it is zero-stable. HereThe consistency conditions hold for every $a$. The two quadratic roots have product one. They are distinct and on the unit circle exactly when $0<a<2$; at $a=2$ the root $-1$ is repeated, while outside this interval one root has modulus greater than one. The root condition for a multistep method therefore givesThe polynomial exactness conditions hold through degree two, while the degree-three residual is $6-a$. Every convergent member thus has(The exceptional value $a=6$ has formal order four but is not zero-stable.)

Solved by gpt-5.6-sol high.