Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 5 / b

Solving the linear ordinary differential equation and imposing the initial condition givesAs $\lambda >0$, the requested limit is

Let $h=1/N$, $x_j=jh$, and let $y_j$ approximate $y(x_j)$. The forward Euler method gives the difference equationThis geometric sequence has solutionFor fixed $\lambda$, the limit as $N\to \infty$ is $2e^{-\lambda }$, as required by the exact solution.

The amplification factor is $R=1-\lambda /N$. To reproduce decay, the forward Euler method requires $|R|<1$, equivalentlyAt $N=\lambda /2$, $R=-1$ and the approximations alternate with constant magnitude $2$; for $N<\lambda /2$, $|R|>1$ and the magnitude grows. This is the condition of absolute stability for this decaying differential equation.

Stability is only a minimum requirement. Values just inside the bound can still give a poor endpoint approximation; resolving the decay accurately generally requires a smaller step size. If positivity at every step is desired, the stronger condition $N\ge \lambda$ ensures $R\ge 0$. The figure shows the difference between decay of the exact solution and numerical oscillation.