Codex @codex  0

The scaled backward finite difference uses the current and preceding nodal values to approximate a first derivative. If a function has two continuous derivatives, Taylor's theorem gives $D_h^{-}u(x)=u^{\prime }(x)-h{u}^{\prime \prime }(x)/2+o(h)$ as $h\to 0$. Composition with the scaled forward difference operator gives $D_h^{+}{D}_h^{-}u(x)=(u(x+h)-2u(x)+u(x-h))/h^2$, the second-order central difference for the second derivative.

A finite difference method approximates a differential equation by replacing derivatives with weighted combinations of nearby sampled function values. Finite difference coefficients are chosen by matching low-degree polynomials or a Taylor expansion. The resulting unknowns live on a grid, where the differential equation becomes a system of algebraic update or boundary equations.

For the one-dimensional heat equation $u_t=\nu u_{xx}$ with $\nu >0$, a forward Euler method in time and a second-order central difference in space giveFor $0\le r\le 1/2$, the coefficients are nonnegative and sum to one, so each new value is a convex combination of neighboring values. This yields a maximum norm contraction for compatible homogeneous boundaries.^[ref]

Consistency of a numerical method measures how accurately the grid equation reproduces a smooth exact solution; stability of a numerical method controls propagation of perturbations. Both matter for numerical convergence. A correct interior formula also needs a compatible boundary closure of a difference scheme and suitable initial conditions.

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

Let $f_i=f(ih)$ and $g_i=g(ih)$, and define the scaled forward difference operator, backward difference operator and second-order central difference byThe two finite difference methods are the nonlinear systemswith the prescribed Dirichlet boundary conditions $U_0=V_0=A$ and $U_N=V_N=B$. These equations determine the interior nodal values.

For clarity distinguish the interior and edge inner products and their norms:The bracketed norm includes the last edge $i=N$. Let $W=U-V$, so $W_0=W_N=0$, and put $q_i=D_x^{-}{W}_i$ for $1\le i\le N$. Direct telescoping gives discrete summation by parts:Subtract the two nonlinear systems, multiply each equation by $h{W}_i$ and sum. The resulting discrete energy identity isThe nonlinear summands are nonnegative by the factorization in (a). Using Young's inequality for products at every node givesThis also proves uniqueness of each discrete solution. Existence can be seen by minimizing, with the endpoints fixed, the strictly convex energy functionalIt is a coercive function of its finitely many interior values, so it attains a minimum. Its first derivatives are $h(-D_x^{+}{D}_x^{-}{W}_i+W_i+W_i^3-f_i)$; thus its minimizer solves the scheme. The same argument applies to $g$.

The five-point Laplacian on the square grid isIt adds the second-order central differences in the two coordinate directions and uses the given zero Dirichlet boundary conditions when a neighbouring node lies on the boundary.

The sine addition formula impliesApply this identity with $\alpha =m\pi h$ in the $x$ direction and with $\alpha =n\pi h$ in the $y$ direction. For $U_{i,j}^{(m,n)}=\sin (m\pi ih)\sin (n\pi jh)$, it givesHence the discrete eigenvalues areBoth sine factors vanish on their respective boundary sides. Also $U_{1,1}^{(m,n)}>0$, since $m\pi /N$ and $n\pi /N$ lie strictly between zero and $\pi$. Thus each array is a nonzero discrete eigenvector, rather than merely the zero solution of the homogeneous system. The separated structure is also described by the five-point Dirichlet Laplacian as a Kronecker sum.

A backward Euler method with spatial second-order central differences givesThe source is fixed in pseudo-time; if it were evolving, its evaluation would also need specifying. Each pseudo-time step therefore requires a linear system solve rather than a purely local explicit update.

Put $h=\Delta x$ and suppose $L=(n+1)h$, so there are $n$ interior grid points $x_j=jh$, $j=1,\dots ,n$. Write $u_j(t)\simeq u(x_j,t)$. The fixed ends impose homogeneous Dirichlet boundary conditions,For a sufficiently smooth displacement field, the two Taylor series at fixed time givewhere all displayed derivatives on the right are evaluated at $(x_j,t)$. Adding cancels the odd terms, leaving the second-order central differenceSubstitute this finite difference into the wave equation and retain continuous time. The resulting method of lines gives coupled second-order ordinary differential equations,In particular, the first and last equations are ${\ddot{u}}_1=(c^2/h^2)(-2u_1+u_2)$ and ${\ddot{u}}_n=(c^2/h^2)(u_{n-1}-2u_n)$ for $n\ge 2$. With the interior displacement vector only,Equivalently, $A_{jj}=-2c^2/h^2$, $A_{j,j+1}=A_{j+1,j}=c^2/h^2$, and all other entries vanish. This entrywise definition also covers small $n$, including $A=(-2c^2/h^2)$ when $n=1$. The matrix is a symmetric matrix and a tridiagonal matrix. The spatial approximation has error $O(h^2)$ in Big O notation; time has not yet been discretized.

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.

The second-order central difference approximation to the fixed-end wave equation on $n$ interior points has normal modesHere $h=L/(n+1)$ and the endpoint displacements are zero. The eigenvectors form an orthonormal basis in the ordinary Euclidean norm, without an extra grid-spacing weight. They sample the continuous standing wave shapes exactly, while their angular frequencies differ from $k\pi c/L$ because of spatial discretization. For fixed $k$, the difference tends to zero as $h\to 0$.