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Use von Neumann stability analysis on the infinite grid. For the Fourier mode $u_{m,k}=\hat{u}(t)e^{i(m\xi +k\eta )}$, the semidiscrete eigenvalue isIts real part is nonpositive for every frequency. The mode evolves by $\hat{u}(t)=e^{t{\lambda }_h}\hat{u}(0)$, whose modulus does not grow. The discrete Fourier transform and Parseval identity therefore yieldwith a bound independent of $h$, where $\Vert u{\Vert }_{{\ell }_h^2}^2=h^2{\sum }_{m,k}|u_{m,k}{|}^2$. Hence the semidiscrete method is stable. Frequencies with $\eta =0$ are nondissipative rather than unstable.

The same sign can be seen directly by summation by parts: the centred $x$ operator is skew-adjoint, and for the unitary $y$ shift $S_y$,This is an energy version of the dissipative second-order forward advection stencil. A subsequent time integrator would need its own stability condition; semidiscrete stability does not assert that every fully discrete method is stable.

Eigenvalue analysis proves mesh-uniform stability when it is accompanied by control of eigenvectors or an energy norm; the signs of eigenvalues alone are insufficient. For an evolution equation, the method of lines first produces $u_h^{\prime }=L_h{u}_h$. A time integrator then gives an amplification matrix $S_{h,\delta }$, for example $S_{h,\delta }=R(\delta L_h)$ for a Runge-Kutta method. The useful stability requirement on a fixed physical time interval iswhere $C_T$ is independent of the refining spatial and temporal meshes. The semidiscrete analogue is $\Vert e^{t{L}_h}{\Vert }_h\le C_T$ for $0\le t\le T$. These are bounds on propagation of initial errors and residuals, not merely statements that each finite matrix has a bounded solution over its own finite time interval.

If $L_h$ is a normal matrix in the chosen discrete inner product, it has a unitary eigenbasis. ThenFor dissipative problems it is sufficient that every $\delta {\lambda }_j$ lie in the integrator's linear stability domain. A diagonalizable family also suffices if its eigenvector matrices have uniformly bounded condition numbers: the same maximum is multiplied by $\Vert V_h\Vert \Vert V_h^{-1}\Vert$. Without that uniform bound the argument does not establish mesh stability. For finite-time estimates, an amplification modulus $1+O(\delta )$ can also be acceptable; strict contraction at every step is a stronger property.

For a constant-coefficient stencil on the full lattice or a periodic grid, von Neumann stability analysis provides the eigenvectors explicitly. A Fourier mode diagonalizes each translation-invariant spatial difference, giving a scalar symbol ${\lambda }_h(\theta )$ or a small matrix symbol for a system. The discrete Fourier transform and Parseval identity turn a uniform modal bound into an actual discrete L2 norm bound. The two-dimensional stencil in question 3 has $\mathrm{Re}{\lambda }_h=-(1-\cos \eta {)}^2/h\le 0$, so its continuous-time amplification is contractive; a later time integrator must still be checked. With boundaries one must analyze the boundary rows as well: a whole-line Fourier symbol cannot detect an unstable boundary closure.

For the heat equation $u_t=\kappa u_{xx}$ on a unit interval with zero Dirichlet data, the centred second difference has a sine eigenbasis and eigenvaluesExplicit Euler method multiplies mode $j$ by $1+\delta {\lambda }_j$. Requiring all factors to have modulus at most one yields $\kappa \delta /h^2\le 1/[2{\sin }^2((N-1)\pi /(2N))]$ on a given grid, and the sharp grid-independent sufficient limit isThese parabolic time-step restrictions express the growth of the largest-magnitude diffusion eigenvalue like $h^{-2}$. Backward Euler method instead has factor $(1-\delta {\lambda }_j{)}^{-1}$ and the Crank-Nicolson method factor $(1+\delta {\lambda }_j/2)/(1-\delta {\lambda }_j/2)$, both bounded by one for every $\delta \ge 0$. Their unconditional stability follows from A-stability, with much stronger damping of the shortest scales for backward Euler.

For $u_t=-a{u}_x$, $a>0$, explicit time stepping with a backward spatial difference givesThus the upwind finite difference scheme is contractive exactly for $0\le \nu \le 1$, a Courant–Friedrichs–Lewy condition matching the direction and speed of information propagation. A centred spatial difference has purely imaginary eigenvalues, and forward Euler gives $|S{|}^2=1+{\nu }^2{\sin }^2\theta >1$. Under a fixed positive hyperbolic Courant number, high-frequency data grow by a fixed factor in each of $O(1/h)$ steps, proving instability as the mesh is refined. This is not a claim that no exceptionally small coupled step can ever give a finite-time bound: if $\delta =O(h^2)$, the excess per step is only $O(\delta )$, but the usual hyperbolic scaling is unstable. Higher-order explicit integrators can possess nontrivial imaginary-axis stability intervals and are then suitable for skew-adjoint advection or the energy-form first-order wave equation.

Variable coefficients usually prevent Fourier diagonalization, but may preserve a self-adjoint spatial operator. The positive-face diffusion discretization in question 5 is an example. Its discrete energy identity makes $A+B$ symmetric negative definite, so a unitary eigenbasis still exists even without a closed eigenvalue formula. More generally, a dissipative operator $L_h$ in a discrete energy inner product obeyswhich directly proves a uniform semigroup estimate. The dissipative Cayley-transform contraction also follows without normality: for $\alpha \ge 0$,Taking $v=(I-\alpha L_h{)}^{-1}u$ proves contraction of the Crank–Nicolson factor. The contractivity of split Crank-Nicolson diffusion then follows from multiplying two contraction bounds; simultaneous eigenvectors of the two directional matrices are unnecessary.

The negative spectra do not imply uniform semidiscrete stability example shows what can go wrong. LetBoth eigenvalues are $-1$ for every mesh, but applying the propagator to the second unit vector at $t=1$ gives norm at least $1/(eh)$. The evolution is therefore not mesh-uniformly stable. For a discrete matrix, a unit-modulus eigenvalue with a nontrivial Jordan block gives powers growing polynomially in the step number. Even strictly interior eigenvalues can allow large transient growth when eigenvectors are badly conditioned. A pseudospectrum or numerical range of an operator, and especially a direct energy estimate, can reveal behavior missed by eigenvalues.

A resolvent estimate provides another way to see the needed uniformity. If $\Vert S_h^n\Vert \le C$ for every $n$, then for $|z|>1$,This shows directly that a uniform power estimate imposes more than the location of the spectrum. Conversely, resolvent-to-power estimates in varying dimension must themselves be uniform; a bound with constants depending on the number of grid points is not enough.

For a linear multistep method, each spatial eigenvalue gives an amplification polynomial, and all roots must satisfy the root condition for a multistep method, including simplicity of any unit-modulus root. Merely following the root approximating $e^{\delta \lambda }$ misses parasitic modes. The temporal and spatial analyses must therefore be combined with the appropriate zero-stability and uniform modal bounds.

Finally stability connects to convergence. If the exact grid samples have normalized residual ${\tau }^n$ and the error obeys $e^{n+1}=S_h{e}^n+\delta {\tau }^n$, iteration givesA residual tending uniformly to zero then gives convergence on $[0,T]$. For a well-posed linear initial-value problem and a consistent linear finite-difference approximation, this is the forward implication of the Lax equivalence theorem. The practical conclusion is to analyze the full operator including boundaries, select a time method whose stability region contains the relevant scaled spectrum, and verify that eigenvectors or an energy estimate make the resulting bounds uniform as the grid is refined.

For $y=(1,1,-1,-1)$, the even and odd subsequences are both $(1,-1)$ and each length-two inverse discrete Fourier transform is $(0,2)$. Since ${\omega }_4=i$, the FFT butterfly givesFor $y=(1,-1,1,-1)$, the even transform is $(2,0)$ and the odd transform is $(-2,0)$. The same assembly gives

Use the printed unnormalized inverse discrete Fourier transform, with positive exponent: $x_{\ell }={\sum }_j{\omega }_n^{j\ell }{y}_j$. Its natural domain and codomain are ${\mathbb{C}}^n$; the printed real-to-real designation is not literal, as the first example below has complex output. There is no factor $1/n$ in the convention of this question.

For period two in each direction, use Fourier coefficientsIntegration of the periodic Poisson equation gives the necessary compatibility condition $f_0=0$. With it, the normalization fixes $u_0=0$ and every other Fourier mode gives $-{\pi }^2|k{|}^2{u}_k=f_k$. The Fourier-Galerkin approximation is thereforeOne may evaluate the coefficients by a discrete Fourier transform for a pseudospectral implementation, with the corresponding quadrature/aliasing error distinguished from truncation.

Convergence with spectral accuracy means faster decay than every fixed algebraic power of $N$. A smooth periodic extension gives this rate; an analytic periodic extension to a complex strip gives the stronger exponential rate $O(e^{-cN})$ on a slightly narrower strip. Division by $|k{|}^2$ preserves that Fourier decay, so the normalized solution has the same type of spectral convergence.

Two qualifications are required by the literal source assumptions. The mean-zero condition on $f$ was not stated: $f\equiv 1$ meets the printed analyticity and endpoint equalities but has no periodic solution. Also endpoint equality of $f$ alone does not make its periodic extension analytic. For example $f(x,y)=x^2-1/3$ is analytic, has matching endpoint values and zero mean, but its nonzero Fourier coefficients are $f_{(k,0)}=2(-1{)}^k/({\pi }^2{k}^2)$. Then $u_{(k,0)}=-2(-1{)}^k/({\pi }^4{k}^4)$, with only algebraic decay. Thus the standard spectral-speed conclusion is valid under the intended analytic-periodic torus assumption and compatibility, not for every function satisfying only the displayed endpoint values.

Use the discrete norm $\Vert U{\Vert }_h^2=h{\sum }_{m=1}^N|U_m{|}^2$, with periodic indices. Two independent arguments prove uniform stability of a numerical method.

For Fourier stability analysis, a discrete Fourier mode $U_m(t)={\hat{U}}_k(t)e^{im{\theta }_k}$ with ${\theta }_k=2\pi k/N$ has semidiscrete eigenvalueIts real part is nonpositive, so $|{\hat{U}}_k(t)|\le |{\hat{U}}_k(0)|$. The unitary discrete Fourier transform and discrete Parseval identity then give $\Vert U(t){\Vert }_h\le \Vert U(0){\Vert }_h$ for every $N$, $t\ge 0$ and fixed real $\alpha$.

For the energy method, denote the centered first difference by $D_0$ and the forward difference by $D_{+}$. Periodic discrete summation by parts makes $D_0$ skew-adjoint and gives $⟨U,D_{xx}U{⟩}_h=-\Vert D_{+}U{\Vert }_h^2$. ThereforeThe constant mode is preserved rather than strictly damped. Both proofs apply equally to differences of two solutions. The semidiscretization is contractive in the discrete L2 norm. A later choice of a time integrator would require its own stability analysis; no time-step condition is needed for this continuous-time result.

Use the unitary discrete Fourier transform, with $e(x)=\exp (2\pi ix)$:For a unit $a$, substitute $m=an$ in the sum. The multiplicativity of the Dirichlet character gives $\chi (a^{-1}m)=\overline{\chi (a)}\chi (m)$, henceThe complex conjugation is present in the original PDF and lost in the converted TeX. It matters for nonreal characters.

Now let $q=p^k$ and let $\chi$ be primitive. Since it does not descend to $p^{k-1}$, there is a unit $u\equiv 1(\mathrm{mod}{p}^{k-1})$ with $\chi (u)\ne 1$. For $k=1$, reduction is to the unit group modulo one. If $p\mid a$, then $a(u-1)\equiv 0(\mathrm{mod}q)$, so multiplication of the summation variable by $u$ leaves its exponential factor unchanged. It follows that $\hat{\chi }(a)=\chi (u)\hat{\chi }(a)$, and therefore $\hat{\chi }(a)=0$. Also $\chi (a)=0$. This proves the formula at every nonunit as well as every unit, including $a=0$.

Put $n_n=c_n^{†}{c}_n$ and $P_n={\prod }_{m<n}(1-2n_m)$. The Jordan–Wigner transformation is $S_n^{+}=P_n{c}_n$, $S_n^{-}=P_n{c}_n^{†}$ and $S_n^z=1/2-n_n$. Since ${\sigma }_n^x=P_n(c_n+c_n^{†})$, $P_{n+1}=P_n(1-2n_n)$ and $(c_n+c_n^{†})(1-2n_n)=c_n^{†}-c_n$, adjacent bonds becomeThus, apart from the end bond,The end bond contains the global fermion parity and sets the sector-dependent periodic or antiperiodic fermion modes. It contributes an order-one boundary term, which is negligible for the thermodynamic energy density; it is not identically zero for the finite spin chain.

Choose the discrete Fourier transform convention $c_n=N^{-1/2}{\sum }_k{e}^{-ikn}{c}_k$. Hopping gives $-2J\cos k{c}_k^{†}{c}_k$. Opposite-momentum pairing gives $iJ\sin k(c_k^{†}{c}_{-k}^{†}+c_k{c}_{-k})$, using the canonical anticommutation relations to antisymmetrize the coefficient. ThereforeFor the Ising-chain Nambu spinor ${\Psi }_k=(c_k,-i{c}_{-k}^{†}{)}^T$, expansion ofrecovers every term: the diagonal contributes $J(g-\cos k)(n_k+n_{-k}-1)$, and the two off-diagonal entries supply the pairing. Since ${\sum }_k\cos k=0$, the constant is $-JgN$. Reversing the Fourier sign changes the pairing convention; the specified sign makes the displayed matrix agree directly.

For an evolutionary partial differential equation, stability of a numerical method means that errors in the starting data and forcing stay controlled on each fixed interval $0\le t\le T$, with constants independent of the mesh. Fourier stability analysis is especially effective for a constant-coefficient finite difference method on a uniform infinite or periodic grid: translation invariance makes different Fourier modes evolve independently.

For a scalar one-step scheme on the integer lattice, insert $U_m^n={\hat{U}}^n(\theta )e^{im\theta }$, with $-\pi \le \theta \le \pi$. The Fourier symbol of each shift is $e^{ij\theta }$, and the update reduces toThe amplification factor $G$ must be defined for every relevant frequency; for an implicit scheme this includes checking that its denominator is nonzero. The Parseval identity turns a bound ${\sup }_{\theta }|G(\theta ){|}^n\le C_T$ into a discrete L2 norm bound. Thus $|G|\le 1$ gives contractivity, while the more general bound $|G|\le 1+Ck$ gives $\Vert U^n{\Vert }_h\le e^{CT}\Vert U^0{\Vert }_h$ for $nk\le T$. Conversely, frequencies with amplification uniformly greater than one produce unstable wave packets or periodic Fourier modes under refinement.

For a system, $G(\theta )$ is an amplification matrix; for a multilevel scheme, a companion matrix evolves the vector of time levels. The actual requirement is a uniform bound on matrix powers, not just their spectral radius. A nontrivial Jordan block at a unit-modulus eigenvalue creates polynomial growth in the time index. Even simple eigenvalues inside the unit disk can fail to give a uniform bound if the eigenvector matrices become ill-conditioned as the mesh changes. Uniformly controlled diagonalization of a matrix, or a suitable quadratic energy estimate, resolves this issue. The power boundedness of a two-level Fourier scheme illustrates why root multiplicities and conditioning matter.

For the heat equation, put $\mu =k/h_x^2$. Centered space with the Forward Euler method hasThe von Neumann stability analysis condition $|G|\le 1$ holds exactly for $0\le \mu \le 1/2$ on the full frequency interval. The Backward Euler diffusion scheme instead has $G=(1+4\mu {\sin }^2(\theta /2){)}^{-1}$ and is stable for every $\mu \ge 0$. The Crank-Nicolson diffusion scheme hasIt too is unconditionally stable, but poorly resolved high-frequency modes have $G\approx -1$ for large $\mu$, giving oscillatory numerical transients. A-stability therefore does not guarantee strong damping; L-stability distinguishes the damping of the Backward Euler method.

For the advection equation $u_t+c{u}_x=0$, let $\nu =ck/h_x$. Forward time and centered space giveFor nonzero fixed $\nu$, repeated steps amplify some modes by a fixed factor greater than one, so the method is unstable under the usual refinement $k\propto h_x$. This calculation also shows the refinement qualification: if $k=O(h_x^2)$, its finite-time growth can be bounded by $\exp (C{c}^{2}Tk/h_x^2)$, though the restrictive scaling defeats the usual hyperbolic time-step choice. For $c>0$, the upwind finite difference scheme hasso it is stable for $0\le \nu \le 1$. The Lax-Wendroff advection scheme instead hasgiving $|\nu |\le 1$. These examples separate the effects of the spatial stencil, temporal approximation and Courant number; a higher order of a numerical method by itself does not establish stability.

A periodic boundary condition is ideal for this analysis. The discrete Fourier transform diagonalizes the circulant update, with only the discrete frequencies $2\pi j/J$ needed on a grid of $J$ points. A bound on all $\theta$ is a convenient guarantee over every such grid. On the whole line, the Fourier transform uses a continuous frequency interval and the Parseval identity proves the corresponding square-summable-data estimate.

Homogeneous Dirichlet boundary conditions do not admit arbitrary complex exponential Fourier modes. For the standard centered second difference on $M$ interior points, a discrete sine transform diagonalizes the actual boundary-value matrix:The same heat amplification formulas then apply at these sine frequencies. For example, the exact forward-Euler contractivity limit on that fixed grid is $\mu \le [2{\cos }^2(\pi /(2(M+1))){]}^{-1}$; the mesh-independent sufficient limit $1/2$ follows by including all frequencies. Homogeneous Neumann boundary conditions can similarly permit a discrete cosine transform, provided the endpoint discretization and its weighted inner product are chosen consistently. The constant mode is then present, reflecting conservation of the heat equation's spatial mean. For inhomogeneous boundary conditions, subtract a suitable lifting of the prescribed boundary data and estimate the induced source using the homogeneous evolution's bound and a Duhamel principle estimate. Bounds for the lifting and source must themselves be uniform in the mesh.

General boundary conditions require separate boundary stability of a finite-difference method. For an advection equation, incoming data are prescribed at the inflow boundary, while an outflow closure must respect the outgoing characteristics. An interior periodic Fourier symbol cannot detect a growing mode confined near a boundary. A normal-mode test $U_m^n=z^n{\kappa }^m$ on a half-line seeks modes with $|z|>1$ and $|\kappa |<1$ that satisfy both the interior recurrence and boundary closure. Their existence proves instability, and uniform control requires more than merely excluding isolated growing roots; the Uniform Kreiss--Lopatinskii condition addresses boundary resolvent bounds. Alternatively, a direct energy method using summation by parts can include the boundary terms and establish the required estimate.

Variable coefficients and nonuniform meshes usually destroy exact Fourier transform diagonalization. Frozen-coefficient Fourier stability analysis is then a useful diagnostic, but it is not automatically a proof for the full variable-coefficient boundary problem. The mesh-uniform energy method in question 5 is an example of a direct proof. Also, an L2 norm proof is not automatically a mesh-uniform maximum-norm proof: finite-dimensional norm-equivalence constants may grow with the number of grid points.

Finally, stability measures error propagation; consistency of a numerical method measures the defect of inserting the exact solution. For a well-posed linear initial-value problem in the chosen norm, the Lax equivalence theorem says that a consistent approximation is convergent exactly when it is stable. Boundary discretization and starting data must be included in that assertion. A Fourier calculation proves convergence only after consistency, well-posedness and the actual boundary treatment have also been checked.

Apply a discrete Fourier transform, with lattice spacing one:The wavevectors lie in a Brillouin zone, which we choose as $[-\pi ,\pi )$. Pairing $k$ with $-k$ makes the quadratic HamiltonianA bosonic Bogoliubov diagonalization uses $a_k=\cosh {\theta }_k{\alpha }_k-\sinh {\theta }_k{\alpha }_{-k}^{†}$. The canonical commutation relations hold because ${\cosh }^2{\theta }_k-{\sinh }^2{\theta }_k=1$. The anomalous terms vanish when $\tanh 2{\theta }_k=B_k/A=\cos k$, givingNear $k=0$ and $k=\pi$, the dispersion relation is linear: ${\omega }_k\sim 2JS|k-k_0|$. These are the low-energy antiferromagnetic spin waves; the two zeroes are related by the two-sublattice description. The transformation is singular at the exact zero modes, which require an infrared regulator or separate treatment of the collective rotation.

With angular frequency $\lambda$, the periodogram isAt Fourier frequencies $2\pi j/T$ it is the squared modulus of the normalized discrete Fourier transform. If a nonzero mean is unknown, subtract the sample mean first.

For a zero-mean stationary process, expanding the square givesUnder absolute summability of the covariances this converges to the spectral density of a stationary process. Thus the periodogram is generally biased at finite $T$, but asymptotically unbiased under this short-memory time series condition.

It is nevertheless not a pointwise estimator with statistical consistency unless it is smoothed. For Gaussian white noise, at a nonzero Fourier frequency other than the Nyquist frequency, the real and imaginary Fourier components are independent normal variables. Exactly,where $f={\sigma }^2/(2\pi )$. The variance does not decrease with $T$. Under usual short-memory time series assumptions the same exponential limit is asymptotic for general processes. Averaging nearby frequencies or using a lag-window estimator reduces variance; a frequency bandwidth tending to zero while $T$ times that bandwidth tends to infinity can give statistical consistency. The raw plot remains useful for detecting strong periodic peaks, but increasing the record length alone does not remove its pointwise noise.

For a fully discrete linear partial differential equation evolution, write $U^{n+1}=C_{h,k}{U}^n$ in a specified discrete norm. Finite-time stability of a numerical method meanswith $C_T$ independent of the allowed meshes. For a multistep method, augment the state with its time-history values and include a stable starter. Bounds may grow as $e^{\omega T}$ when the continuous problem has growth; demanding decay for all time would be a stronger assertion. The treatment of initial data, forcing, Dirichlet boundary conditions, and the norm is part of the hypothesis, not a detail supplied by an interior calculation.

For constant coefficients on an infinite or periodic uniform grid, von Neumann stability analysis substitutes the Fourier mode $U_j^n=G(\theta {)}^n{e}^{ij\theta }$. The discrete Fourier transform and the Parseval identity turn a uniform multiplier estimate into a discrete L2 norm estimate. In a one-step scalar scheme, $|G(\theta )|\le 1$ proves contractivity, while $|G(\theta )|\le 1+\omega k$ gives finite-time stability. In a multilevel scheme, every root of a polynomial in its amplification polynomial of a multilevel finite difference scheme matters, as does uniform control of the associated companion matrix.

For the Forward Euler diffusion scheme with $\mu =k/h^2$,Thus $|G|\le 1$ for all frequencies exactly whenSufficiency follows since $G\in [-1,1]$; necessity follows from the highest-frequency mode $\theta =\pi$ on even periodic grids. This gives the usual parabolic mesh restriction $k\le h^2/2$. For the Backward Euler diffusion scheme,so every $\mu \ge 0$ is stable. The Crank-Nicolson method hasagain of modulus at most one for all $\mu \ge 0$, but stiff modes approach $-1$ rather than zero. These are PDE manifestations of the A-stability and L-stability distinctions for time integration.

For advection $u_t+a{u}_x=0$, assume $a>0$ and set $\nu =ak/h$. The upwind finite difference scheme is $U_j^{n+1}=(1-\nu )U_j^n+\nu U_{j-1}^n$, withIt is contractive when $0\le \nu \le 1$. This also has a direct maximum-norm proof: each newest value is a convex combination of two old values. The resulting Courant–Friedrichs–Lewy condition expresses that the numerical domain of dependence covers the physical one. For negative $a$, the upwind direction must be reversed. By contrast, Forward Euler method time stepping with the centered first finite difference has $G=1-i\nu \sin \theta$ and $|G|>1$ for nonzero $\nu \sin \theta$. Under fixed nonzero $\nu$ refinement it is unstable, since a fixed nontrivial mode grows geometrically over $T/k$ steps. This is not a claim of instability under every imaginable mesh coupling: $k=O(h^2)$ instead bounds its spurious finite-time growth, since $\log |G|\le a^2{k}^2/(2h^2)$.

The leapfrog advection scheme illustrates a multilevel subtlety. Its amplification polynomial of a multilevel finite difference scheme isFor $|\nu |<1$, both roots of a polynomial have unit modulus and separation at least $2\sqrt{1-{\nu }^2}$. A uniformly conditioned eigenbasis of the companion matrix then proves power boundedness of a two-level Fourier scheme for arbitrary history data. At $|\nu |=1$ and a grid admitting $\theta =\pi /2$, the roots coincide on the unit circle, and a Jordan block produces $n{\xi }^n$ growth. Thus the endpoint fails the ordinary arbitrary-history root condition for a multistep method; checking only that both moduli equal one misses the instability. Restricting the starter or filtering the parasitic mode is an additional hypothesis.

The energy method handles variable coefficients and finite boundaries for which Fourier stability analysis may be unavailable. For the Backward Euler diffusion scheme with homogeneous Dirichlet boundary conditions, define $\Vert U{\Vert }_h^2=h{\sum }_j|U_j{|}^2$. The discrete summation by parts identity isTake the inner product of $U^{n+1}-U^n=k{L}_h{U}^{n+1}$ with $2U^{n+1}$. The elementary identity $2\mathrm{Re}⟨x-y,x⟩=\Vert x{\Vert }^2-\Vert y{\Vert }^2+\Vert x-y{\Vert }^2$ givesThis proves unconditional contractivity including the actual boundary treatment. More generally, a dissipative operator $L_h$ gives $(I-k{L}_h{)}^{-1}$ of operator norm at most one: with $V=(I-k{L}_h)U$, dissipativity implies $\mathrm{Re}⟨V,U⟩\ge \Vert U{\Vert }^2$, and the Cauchy-Schwarz inequality gives $\Vert U\Vert \le \Vert V\Vert$. In finite dimensions this also proves invertibility. For the second-order backward differentiation formula, the BDF2 discrete energy identity used in question 4 controls both time levels and proves unconditional diffusion stability. A repeated root strictly inside the disk is harmless here; the energy argument supplies the uniform bound without a singular eigenvector formula.

A second technique is eigenvalue stability analysis of a finite difference method. Under the method of lines, a time integrator advances $U^{\prime }=L_{h}U$ by $R(k{L}_h)$. If $L_h$ is a normal matrix, the scalar linear stability domain criterion on all $k{\lambda }_j(L_h)$ controls its powers exactly in the corresponding L2 norm. For the centered Dirichlet discrete Laplacian, its eigenvalues lie between $-4/h^2$ and zero; intersecting this interval with a time integrator's stability interval determines its mesh restriction. A uniform bound on the diagonalization of a matrix is needed for nonnormal diagonalizable systems. Eigenvalues alone can be misleading. For example,has both eigenvalues inside the disk for $0<k<1$, but the upper-right entry of $C_{h,k}^n$ is $nk(1-k{)}^{n-1}/h$. Taking $k=h$, $nk\to T>0$, this grows like $T{e}^{-T}/h$. Hence stable scalar eigenvalues do not give mesh-uniform stability. Boundary closures can introduce precisely the extra growth that an interior Fourier symbol overlooks, so boundary stability of a finite-difference method must be checked separately.

For nonlinear spatial discretizations, a useful replacement for Fourier stability analysis is a convexity argument. If the Forward Euler method map $E_k(U)=U+kF(U)$ is nonexpansive in a chosen norm for $k\le k_{\mathrm{FE}}$, the second-order strong stability preserving Runge-Kutta methodis also nonexpansive under the same restriction. Indeed, for two inputs, the triangle inequality gives $\Vert U_{\mathrm{new}}-{\tilde{U}}_{\mathrm{new}}\Vert \le \frac{1}{2}\Vert U-\tilde{U}\Vert +\frac{1}{2}\Vert E_k(Y)-E_k(\tilde{Y})\Vert \le \Vert U-\tilde{U}\Vert$. Its Taylor expansion is $U+kF(U)+k^2{F}^{\prime }(U)F(U)/2+O(k^3)$, proving order two. Such strong stability preserving Runge-Kutta methods transfer suitable forward-step bounds without relying on a linear spectral calculation.

Finally, consistency of a numerical method explains what stability buys. For a linear Hadamard well-posed problem, the Lax equivalence theorem equates convergence of a consistent discretization with its stability, under the stated approximation-space and norm hypotheses. Directly, if the error satisfies $e^{n+1}=C_{h,k}{e}^n+k{\tau }^n$, iteration gives the discrete Duhamel principleThus vanishing normalized local truncation error and initial error yield convergence. With a nonlinear Lipschitz step estimate, the discrete Gronwall inequality plays the same role. A stable but inconsistent stencil, such as the literal defective formulas in questions 3 and 4 under their usual refinement, is not rescued by any amplification bound. The decisive checks are a mesh-uniform evolution bound, a valid boundary closure, and consistency with the actual PDE.

Let $\omega =e^{2\pi i/N}$. Using the conventionthe discrete Fourier transform of the supplied sequence follows from the binomial theorem:The orthogonality of roots of unity givesSince $m,k$ both lie between zero and $N-1$, it follows that

Let $e_0,\dots ,e_{n-1}$ be the standard basis and let the generator $g$ send $e_j$ to $e_{j+1}$ modulo $n$. The constant vectorspans a trivial module. Since $n$ is even, the alternating vectorspans a sign module on which $g$ acts by $-1$.

For each $1\le k<n/2$, define the real discrete Fourier modesThe plane $W_k=\mathrm{span}\{c_k,s_k\}$ is invariant, and $g$ acts on it by rotation through $2\pi k/n$. Since this angle is neither zero nor $\pi$, $W_k$ is irreducible over $\mathbb{R}$. The traces $2\cos (2\pi k/n)$ are distinct for $1\le k<n/2$, so these modules are pairwise nonisomorphic. Orthogonality of the discrete Fourier transform givesThe dimensions sum to $1+1+2(n-2)/2=n$, so this is the whole regular module: two nonisomorphic one-dimensional summands and $(n-2)/2$ nonisomorphic two-dimensional summands.

Split the generator as $iA-iV$. One Strang splitting step isThe two potential half-steps are componentwise multiplications. The circulant matrix $A$ is diagonalized by the discrete Fourier transform; its eigenvalues areThus the middle step consists of a Fast Fourier transform, multiplication of Fourier coefficient $j$ by $e^{ih{\lambda }_j}$, and an inverse transform. Its cost is $O(M\log M)$ and every factor is unitary, so the implementation preserves the discrete norm exactly up to roundoff.

Consider first a scalar constant-coefficient Cauchy problem for a partial differential equation on the whole line. A translation-invariant spatial discretization is a convolution operator, so the discrete Fourier transform turns it into multiplication by its Fourier symbol. This reduction is exact because Fourier modes are the simultaneous generalized eigenfunctions of every translation-invariant stencil.

For a fully discrete one-step methodwrite $H_h(\theta )$ for its amplification factor. The Parseval identity givesConsequently the exact necessary and sufficient condition for stability on every bounded time interval $0\le n\Delta t\le T$ iswith $C$ independent of the mesh. Sufficiency follows directly from Parseval; necessity follows by choosing transformed initial data concentrated where the multiplier is largest. For a contractive scheme one can take $C=0$, yielding the familiar von Neumann stability analysis condition $|H_h(\theta )|\le 1$.

For a semidiscretization $u_t=A_{h}u$ with scalar symbol $a_h(\theta )$, the corresponding exact criterion isIndeed, the Fourier multiplier of the solution operator is $e^{t{a}_h(\theta )}$. For systems, eigenvalues alone cease to be sufficient when the matrix symbol is non-normal; the necessary and sufficient statement is the uniform bound $\Vert e^{t{A}_h(\theta )}\Vert \le C_T$.

On the whole lattice, a finite stencil defines a Laurent operator, whose operator norm is the essential supremum of its symbol. Restricting the same stencil to a half-line gives a Toeplitz operator. The Cauchy symbol condition remains necessary for a stable initial-boundary scheme because data localized far from the boundary behave like the whole-line problem for a finite time. It is not sufficient: the boundary closure can support growing modes or amplify incoming modes even when every interior Fourier mode is stable.

Boundary stability therefore requires a uniform estimate for the forced half-line recurrence. After a Laplace transform in time and a Fourier transform in tangential variables, one solves a normal-direction recurrence. The Uniform Kreiss--Lopatinskii condition requires the boundary equations to determine its decaying roots with a uniformly bounded inverse. In practical terms, one imposes one independent boundary condition for each incoming characteristic or numerical mode and none for outgoing modes. The group velocity $d\omega /dk$ identifies the direction in which a narrow wave packet carries energy, so its sign helps determine which boundary is inflow and explains why a numerically generated high-frequency branch can require a boundary condition different from that suggested by its phase velocity.

For a two-step method, the Fourier substitution produces an amplification polynomial of a multilevel finite difference scheme. Every root must lie in the closed unit disk, and unit-modulus roots must be simple, uniformly in the wavenumber. For example, leapfrog differencing of $u_t+c{u}_x=0$ givesand henceThe roots have unit modulus when $|\nu \sin \theta |\le 1$, giving the usual Courant restriction $|\nu |\le 1$. At the endpoint, a wavenumber with $|\nu \sin \theta |=1$ produces a repeated unit root and violates the uniform root condition; strict $|\nu |<1$ avoids this marginal linear growth. The second root is the familiar oscillatory computational mode, illustrating why a multilevel scheme requires its full amplification polynomial rather than a single multiplier.

After spatial discretization, a linear time-dependent PDE produces an updateStability over $0\le n\Delta t\le T$ requires a bound $\Vert Q_h^n\Vert \le C_T$ independent of the mesh. If $Q_h$ is normal, the spectral theorem for normal operators givesso eigenvalue analysis is decisive. More generally, if $Q_h=X_h{\Lambda }_h{X}_h^{-1}$, thenThus eigenvalues suffice only when the eigenvector condition numbers are uniformly bounded and unit-circle eigenvalues are semisimple. Defective or increasingly nonnormal matrices can have large powers even though every eigenvalue lies in the unit disk.