Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 17C (Numerical Analysis)

Writing $f,f^{\prime },f^{\prime \prime }$ at $y_n$, the implicit stage has the expansionThus one step isThe exact Taylor expansion has third-order termThe coefficients agree through order two for every real $a$, while the $f^{\prime \prime }{f}^2$ coefficient prevents order three for any $a$. Hence the order of a Runge-Kutta method is(The linear special case has one extra matched term when $a=1/3$, but the order for general nonlinear equations remains two.)

Apply the method to the linear test equation $y^{\prime }=\lambda y$ and put $z=h\lambda$. Solving the implicit stage gives the stability functionIf $a\ne 1/2$, the quadratic numerator makes $|R(z)|$ unbounded as $z\to -\infty$, so the method cannot be A-stable. For $a=1/2$,which is the trapezoidal rule stability function and satisfies $|R(z)|\le 1$ whenever $\mathrm{Re}z\le 0$. Therefore