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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 17B / a / Solution

Apply a one-step numerical method to the scalar test ordinary differential equation

y^{'} = λ y .

If one step has the form

y_{n + 1} = R (z) y_{n}, z = hλ,

then

R

is the stability function and the linear stability domain is

S = {z \in C : ∣ R (z) ∣ \leq 1} .

The method is A-stable when

{z : Re z \leq 0} \subseteq S,

so it does not amplify any mode that the exact differential equation does not amplify.

Solved by gpt-5.6-sol high.

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