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Set $F(h)=f(a+h)+f(a-h)-2f(a)$ and $G(h)=h^2$. Both vanish at zero. For $h\ne 0$, their derivative ratio isTwice differentiability at $a$ makes both quotients tend to $f^{\prime \prime }(a)$. Apply the endpoint L'Hôpital's rule proved in part (iii) for $h\downarrow 0$. For the other side use the reflected positive variable $t=-h$; $F$ and $G$ are even, so their quotient is unchanged. Thus the two-sided limit isThis is the second-order central difference limit. The proof only uses differentiability of $f^{\prime }$ at $a$ and differentiability nearby; it does not incorrectly assume $f^{\prime \prime }$ is continuous.

The original PDF has a spatial shift $u_{m+2}^{n+2}$, so the printed scheme is not the usual second-order BDF discretization. The PDF does have $-2u_m^{n+2}$ on the right; the TeX aid loses that factor two, and that transcription must not be used.

Let $h=\Delta x$, $k=\Delta t$, and expand an exact smooth heat equation solution about $(x_m,t_{n+2})$. Relative to the standard unshifted backward differentiation formula, the printed newest value addsDividing the raw residual by $2k/3$, its expansion isIn particular, for the usual refinement $k=\mu h^2$ with fixed $\mu >0$, the term $3u_x/(\mu h)$ generally diverges. A concrete smooth counterexample is $u=e^{-{\pi }^{2}t/4}\cos (\pi x/2)$, which satisfies the Dirichlet boundary conditions and has $u_x\ne 0$ at $x=1/2$. Thus the printed scheme has no consistent order under fixed-$\mu$ refinement. Its normalized shift error is $O(h/k)$, so one could only investigate a special coupled limit such as $h=o(k)$ after specifying a valid boundary closure; it has no standard joint second-order accuracy.

For the likely intended correction $u_m^{n+2}$, the extra shift terms disappear. The normalized local truncation error isWith a stable, suitably accurate starter, the corrected backward differentiation formula combined with the second-order central difference therefore hasThis is a correction to the printed method, not a claim about it.

Let ${\delta }_x^2{u}_m=u_{m-1}-2u_m+u_{m+1}$. For an exact smooth solution of the diffusion equation, the second-order central difference givesSince the Courant number $\mu =k/h^2$ is fixed, $h^2=O(k)$. Taylor expansion in time givesUsing $u_t=u_{xx}$ and $u_{tt}=u_{xxt}$ cancels the leading terms. Because $k{h}^2=O(k^2)$, the residual, and hence the local truncation error in the convention of the question, is

Let $U_m^n=u(mh,nk)$ be the sampled exact solution and let $A_h$ denote the update matrix of the Forward Euler diffusion scheme. For $0\le \mu \le 1/2$,is a convex combination, with the homogeneous boundary values included. ThereforeThis is the parabolic discrete maximum principle and proves max-norm stability directly.

The Taylor theorem and the second-order central difference give the exact-grid residualfor a sufficiently smooth solution on a fixed interval $0\le t\le T$. If $e^n=U^n-u^n$, thenIteration for $nk\le T$ yieldsThus $\mu \le 1/2$ implies convergence when the initial grid values converge to the initial data.

Here $\Delta x=1/M$ and periodic indexing is taken modulo $2M$. The second-order central difference matrix is the real symmetric circulant matrixSince $V$ is real diagonal, $H=A-V$ is Hermitian. Therefore $iH$ is skew-Hermitian, and

The Haar wavelet provides an explicit one-vanishing moment comparison:It cancels a constant polynomial, but not a nonconstant affine function. To obtain two vanishing moments by an equally transparent construction, take three adjacent unit intervals with heights proportional to $1,-2,1$:The L2 norm normalization follows from $1^2+(-2{)}^2+1^2=6$. Direct integration givesThus $\eta$ has exactly two vanishing moments. In fact its coefficient is a finite difference of adjacent local integrals:The weights are those of the second-order central difference. The first moment balances the centres of the three intervals; it is not enough merely to balance their signed areas.

This $\eta$ is a normalized analyzing function, with no orthonormal basis assertion. Indeed, its unit translate has nonzero L2 inner product:Hence these translates already fail orthogonality. The Daubechies wavelet construction imposes additional refinement and orthogonality requirements; two vanishing moments and unit L2 norm by themselves do not supply those requirements.