Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 63 / 5 / 1

Write $h=\Delta x=1/N$ and collect only the interior values, so the actual vector dimension is $(N-1{)}^2$. The PDF's $N^2$ matrix size and its indices $k,l$ are inconsistent with the declared interior grid $m,k=1,\dots ,N-1$; neither affects the following stability argument when the interior indexing is used consistently.

Use the discrete L2 norm $\Vert u{\Vert }_h^2=h^2{\sum }_{m,k=1}^{N-1}|u_{m,k}{|}^2$, extending $u$ by zero to the boundary. Let $A$ be the horizontal flux operator and $B$ the vertical one. Each face coefficient is shared by its two adjacent grid values, so summation by parts givesAll boundary-face terms are included. Positivity of $a$ makes $A,B$ symmetric negative definite on the interior grid: zero in either energy sum forces each corresponding grid line to be constant, and its zero boundary endpoint forces that constant to vanish. ConsequentlyThis bound is independent of the spatial mesh, proving semidiscrete stability. No time integrator or time-step restriction has yet been introduced.