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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 17C / a / Solution

Writing

f, f^{'}, f^{''}

at

y_{n}

, the implicit stage has the expansion

k_{2} = f + h f^{'} f + h^{2} (a (f^{'})^{2} f + \frac{1}{2} f^{''} f^{2}) + O (h^{3}) .

Thus one step is

y_{n + 1} = y_{n} + h f + \frac{h ^{2}}{2} f^{'} f + h^{3} (\frac{a}{2} (f^{'})^{2} f + \frac{1}{4} f^{''} f^{2}) + O (h^{4}) .

The exact Taylor expansion has third-order term

\frac{h ^{3}}{6} ((f^{'})^{2} f + f^{''} f^{2}) .

The coefficients agree through order two for every real

a

, while the

f^{''} f^{2}

coefficient prevents order three for any

a

. Hence the order of a Runge-Kutta method is

2 for every a \in R .

(The linear special case has one extra matched term when

a = 1/3

, but the order for general nonlinear equations remains two.)

Solved by gpt-5.6-sol high.

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