= Solution
Sample the diffusion coefficient at edge midpoints and write
$$
\begin{aligned}
\alpha&=a((i-\tfrac12)h,jh),&
\beta&=a((i+\tfrac12)h,jh),\\
\gamma&=a(ih,(j-\tfrac12)h),&
\delta&=a(ih,(j+\tfrac12)h).
\end{aligned}
$$
Centered differences of the two fluxes give the conservative <finite difference method>[finite-difference] stencil
$$
\boxed{
\frac1{h^2}\left[
\alpha u_{i-1,j}+\beta u_{i+1,j}
+\gamma u_{i,j-1}+\delta u_{i,j+1}
-(\alpha+\beta+\gamma+\delta)u_{i,j}
\right]}.
$$
For example, the $x$ part is
$$
\frac1h\left[
a_{i+1/2,j}\frac{u_{i+1,j}-u_{i,j}}h
-a_{i-1/2,j}\frac{u_{i,j}-u_{i-1,j}}h
\right].
$$
Taylor expansion about $(ih,jh)$ shows that the odd powers cancel between the two face fluxes and that the result equals $\partial_x(a\,u_x)+O(h^2)$. The same calculation in $y$ gives total truncation error
$$
\boxed{O(h^2)}.
$$
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