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Write $h=\Delta x$, $k=\Delta t$, and assume $a,u$ are sufficiently smooth for the following Taylor expansions. The symmetric half-grid diffusion consistency calculation giveswhereThe normalized local truncation error of the full update is consequentlyThe PDF prints an advection equation, $u_t=a(x)u_x$, whereas this stencil approximates conservative diffusion, $u_t=(a{u}_x{)}_x$. For the literal printed equation the leading defect is $(a-a^{\prime })u_x-a{u}_{xx}$, which generally does not vanish: the method is inconsistent, with an $O(1)$ normalized defect.

For the intended variable-coefficient conservative diffusion equation, the first two terms cancel and ${\mathcal{T}}_{h,k}=O(k+h^2)$. Under the parabolic step scaling $k=O(h^2)$, this is $O(h^2)$, while the unnormalized one-step local truncation error is $k{\mathcal{T}}_{h,k}=O(h^4)$. These two orders use different normalizations. With numerical stability and compatible smooth initial and boundary conditions, the intended diffusion problem has global error $O(k+h^2)$ on a fixed time interval. Positivity and boundedness of $a$ alone do not justify the displayed Taylor expansion; coefficient regularity is also needed for that error formula.