Past exam of the numerical solutions of partial differential equations course of the University of Oxford / 2023 / 3 / d / Solution

Using the semidiscrete Fourier transform from (b), the explicit scheme has amplification factorSet $r=a\mu$ and $s={\sin }^2(\theta /2)\in [0,1]$, so ${\sin }^2\theta =4s(1-s)$. Direct expansion givesThe expression in square brackets is an affine function of $s$. At the two endpoints its values areusing precisely ${\nu }^2\le 2r\le 1$. Every intermediate value is a convex combination of these endpoints, so $|G_E{|}^2\le 1$ for every frequency. The Parseval identity provesThus the scheme is practically stable under the given condition. For $a>0$ it is equivalent to $\Delta t\le 2a$ and $\Delta t\le (\Delta x{)}^2/(2a)$. This is the Forward Euler stability for centered advection-diffusion calculation with the diffusion coefficient retained. The proof does not require all three stencil weights to be nonnegative.