Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 69 / 1

Apply the Runge-Kutta method to the Dahlquist test equation and put $z=h\lambda$. Its stability function isSince $A$ is an invertible matrix, $Q$ has polynomial degree $s$. The adjugate matrix formula gives $\mathrm{deg}P\le s$, but the additional condition givesso actually $\mathrm{deg}P\le s-1$. The specified order of a numerical method implies $R(z)=e^z+O(z^{2s})$ near zero. Therefore the Padé identification of a Radau stability function makes $R$ the $[s-1/s]$ Padé approximant to the exponential function: its numerator and denominator have the required degrees, and its Taylor series agrees through degree $2s-1$.

In the permitted A-stability of near-diagonal exponential Padé approximants, take $m=s-1$, $n=s$. Then $n-2\le m\le n$, so the scheme is A-stable. The same stability function also tends to zero at infinity, giving L-stability.

For one stage, first-order consistency of a numerical method gives $b_1=1$. The condition $b^T{A}^{-1}1=1$ then gives $a_{11}=1$. Hence the scheme is the Backward Euler method:It is a one-node collocation Runge-Kutta method with $c_1=1$. Specifically, choose a linear polynomial $p(t)$ on $[t_n,t_n+h]$ with $p(t_n)=y_n$ and impose $p^{\prime }(t_n+h)=f(t_n+h,p(t_n+h))$. Its derivative is constant, so $p(t_n+h)-p(t_n)=h{p}^{\prime }(t_n+h)$, exactly the Backward Euler method update.