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The discrete L2 inner product is the cell-volume-weighted Hermitian inner product on a uniform grid. Its associated norm is the discrete L2 norm. It provides the finite-dimensional analogue of spatial integration by parts and continuous energy method identities.

For a real smooth solution of the long-wave convection equation with broken Boussinesq symmetry, assume a periodic spatial average or an existing long-interval average with vanishing endpoint fluxes. Integration by parts gives the exact energy method identityThus $\mu <{\mu }_E$ prevents growth of the mean-square temperature at arbitrary amplitude in this averaging class. This sufficient nonlinear bound need not equal the linear instability threshold, and it does not imply pointwise monotonicity of the temperature.

The front velocity is the propagation speed of the earliest boundary of a newly introduced disturbance's support. It asks when an initially unaffected observer can first respond, whereas group velocity asks how a narrow wave envelope translates. A detector threshold defines a practical arrival time, but can miss arbitrarily small earlier responses; a packet peak is another distinct timing convention. In a causal dielectric model, a high-angular frequency contour argument gives exactly zero field for $t<x/c$, even when phase velocity exceeds $c$. This is a statement about that model's relativistic causality, not a universal identification of group velocity with front velocity.^[ref]

For the stipulated Klein-Gordon equation with compactly supported real Cauchy data, front velocity is bounded by $c$, and $c$ is the model's maximal speed of influence. A short energy method outline usesIntegration outside an expanding interval shows that zero initial energy there remains zero. Thus initial support in $[-a,a]$ cannot influence $x>a$ before $t=(x-a)/c$. The positive energy density includes the restoring term ${\omega }_0^2{u}^2/2$: this adds oscillation behind the front without extending its domain of dependence. A particular disturbance may fail to excite the leading boundary, so the maximal speed is not a promise of a detectable response for every source.

A Gaussian function is nonzero arbitrarily far away, so the Gaussian envelope in a Klein-Gordon medium has no sharp initial support boundary. Its small distant tails are already in the initial data. Likewise, keeping only a narrow interval of the frequency spectrum cannot reproduce an exact abrupt onset. Cutting off a Gaussian tail changes its Fourier transform and the accompanying initial velocity; it is a different initial-value problem. The exponential initial-velocity tail for the Klein-Gordon equation is another concrete example of a pre-existing tail outside the main disturbance.

The water-tank measurement in phase velocity records free surface elevation with calibrated resistance probes and uses a ruler and stopwatch for an independent check. The inspected protocol calls the front or back of a group a timing feature. That visible edge establishes an envelope transit estimate, not the first nonzero disturbance in the sense of front velocity.^[ref] A proposed additional control would repeat the transit fit at several detection thresholds and train durations, using matched probe calibration and the same sampling rate. Dependence on the threshold diagnoses wave envelope reshaping or limited sensitivity; records should be compared before reflections return. Such a test could expose limitations of a group velocity estimate, but would not establish an exact causal front or validate the stipulated Klein-Gordon equation for water waves.

The Kronecker sum of square matrices combines their Kronecker products with identity matrices. If $Ax=\lambda x$ and $By=\mu y$, then $x\otimes y$ is an eigenvector with eigenvalue $\lambda +\mu$. Tensor-product spatial grids therefore turn separated coordinate Laplacians into a Kronecker sum. Two Hermitian matrices give a Hermitian Kronecker sum, allowing an energy method without computing every eigenvector.

A reduced long-wave convection model can combine linear damping, destabilizing second derivatives, stabilizing fourth derivatives and gradient nonlinearities. In the displayed normalization the quadratic coefficient $s$ breaks Boussinesq symmetry, while the cubic gradient term dissipates the mean-square temperature. The linear dispersion relation is $\lambda (k)=-1+\mu k^2-k^4$, with critical wavenumber $k=1$ at $\mu =2$. Smoothness and spatial averaging hypotheses must be specified before applying the energy method on an infinite interval.

Fourier stability analysis exploits translation invariance of a linear constant-coefficient finite difference method. On an infinite uniform grid of spacing $h$, a spatial shift acts on $e^{ij\theta }$ by multiplication by $e^{i\theta }$. Hence a scalar two-level scheme with new- and old-time symbols $L(\theta ),N(\theta )$ propagates each Fourier mode by $G(\theta )=N(\theta )/L(\theta )$, provided the new-time operator is invertible. An explicit stencil $U_j^{n+1}={\sum }_{\ell }{a}_{\ell }{U}_{j+\ell }^n$ simply has $G(\theta )={\sum }_{\ell }{a}_{\ell }{e}^{i\ell \theta }$. This converts a spatial evolution into scalar algebra at each frequency.

The Plancherel theorem gives the full norm estimate, rather than just a heuristic about waves. For the discrete Fourier transform $\hat{U}(\theta )={\sum }_j{U}_j{e}^{-ij\theta }$, interpreted in the $L^2$ sense if necessary,Thus a scalar mode bound gives an exact discrete L2 norm bound. Stability on a fixed physical interval requires constants independent of the refining spatial and time meshes:The contraction condition $|G|\le 1$ is sufficient and customary for equations whose exact solutions do not grow. More generally $|G|\le 1+Ck$ suffices, since $(1+Ck{)}^n\le e^{CT}$. At a fixed Courant number, when $G$ is independent of $k$, a mode with modulus strictly greater than one violates the estimate as $n\sim T/k\to \infty$. One must keep this scaling: physical growth such as $e^{\kappa t}$ does not by itself mean finite-time numerical instability. This is finite-time stability versus power boundedness.

For systems, the symbol is an amplification matrix. The requirement is a uniform bound on all its powers, not merely its eigenvalues. A unit-modulus Jordan block $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ has powers $\left(\begin{array}{cc}1& n\\ 0& 1\end{array}\right)$ and is unstable under refinement. A uniformly conditioned eigenbasis or an energy estimate can justify a modal conclusion; eigenvalues alone cannot control a general nonnormal matrix. Multilevel schemes likewise produce an amplification polynomial of a multilevel finite difference scheme, with a uniform root condition and control of starting errors. Repeated unit-modulus roots can give polynomial growth, and roots approaching each other require attention to mesh-uniform constants.

Several elementary examples show how the calculation works. For $u_t+c{u}_x=0$ with $c>0$, forward Euler with centered spatial differences gives $G=1-i\mu \sin \theta$, where $\mu =ck/h$. Since $|G{|}^2=1+{\mu }^2{\sin }^2\theta$, it is unstable at every fixed nonzero Courant number. The upwind finite difference scheme instead gives $G=1-\mu +\mu e^{-i\theta }$ andso its exact contraction range is $0\le \mu \le 1$. The Lax-Wendroff advection scheme has $G=1-i\mu \sin \theta +{\mu }^2(\cos \theta -1)$; expansion gives $|G{|}^2=1-4{\mu }^2(1-{\mu }^2){\sin }^4(\theta /2)$, hence $|\mu |\le 1$. The Courant–Friedrichs–Lewy condition relates these restrictions to the numerical domain of dependence, but is a necessary geometric check rather than a complete stability proof. Implicit schemes can have a global spatial dependence and different Courant ranges, as Question 3 illustrates.

For the heat equation $u_t=\nu u_{xx}$, put $r=\nu k/h^2$. Central spatial differences with forward Euler give $G=1-4r{\sin }^2(\theta /2)$, so $0\le r\le 1/2$. With backward Euler the factor is $[1+4r{\sin }^2(\theta /2){]}^{-1}$, a contraction for every $r\ge 0$. For the Crank-Nicolson diffusion scheme it is $[1-2r{\sin }^2(\theta /2)]/[1+2r{\sin }^2(\theta /2)]$, again a contraction for all $r\ge 0$. At large $r$, however, high-frequency factors approach minus one rather than zero: unconditional stability does not imply strong damping or accurate resolution of every mode.

The connection with convergence is the Lax equivalence theorem for a consistent approximation to a well-posed linear evolution problem. Its forward implication follows directly from the error recursion. If $e^{n+1}=G_h{e}^n+k{\tau }^n$ and $\Vert G_h^j\Vert \le C_T$ for $jk\le T$, thenA convergent initialization and a vanishing normalized local truncation error therefore give convergence. Smooth-data consistency can be extended by density when the approximation and initialization are uniformly bounded. For rough $L^2$ data, cell averages or another L2-compatible initialization of grid data are preferable to undefined point samples. A high formal order does not rescue an unstable amplification operator.

Boundary conditions determine whether Fourier diagonalization is exact. On a periodic grid the update is a circulant matrix, and the finite discrete Fourier transform diagonalizes it at the frequencies $2\pi \ell /N$. Here an interior symbol controls the entire scheme, including the wrapping boundary. On the whole line, the continuous frequency transform serves the same purpose and there is no boundary closure to check.

For a finite interval with homogeneous Dirichlet boundary conditions, centered diffusion is a symmetric tridiagonal problem diagonalized by discrete sine modes. One tests the same diffusion symbol at the admitted sine frequencies, not at arbitrary periodic modes. For homogeneous Neumann boundary conditions, a suitable reflected closure uses discrete cosine modes; the constant mode is present and is neutral for pure diffusion. With the standard reflected endpoint stencil, the natural norm uses half endpoint weights, in which the matrix is self-adjoint. Consequently stable diffusion need not make every solution tend to zero under Neumann conditions. These special orthogonal bases supply an exact substitute for Fourier modes when the operator and closure have the required symmetry.

For transport, prescribe only the incoming physical boundary data. With $u_t+c{u}_x=0$, $c>0$, the inflow is the left endpoint; for $u_t=u_x$ it is the right endpoint. Imposing unrelated data at both ends generally overdetermines the first-order PDE. Outflow rows or ghost points must be chosen compatibly with the interior method. A general boundary closure of a difference scheme destroys the circulant structure, so whole-line von Neumann stability analysis is necessary in many settings but not sufficient for the full initial-boundary problem. A defective boundary row that doubles one boundary unknown each step can create growth even beside a stable interior stencil.

A more refined boundary calculation seeks half-line modes $U_j^n=w^n{\zeta }^j$ with $|\zeta |<1$; these are localized near the boundary. Insert them into both the interior recurrence and boundary equations. A compatible mode with $|w|>1$ proves boundary instability. Absence of one isolated growing mode is not the whole theorem: weakly controlled limiting modes can still destroy a uniform estimate. The Uniform Kreiss--Lopatinskii condition requires a uniformly bounded inverse boundary system on the incoming/decaying mode space. Direct energy estimates are often a clearer alternative to solving that transformed boundary problem.

For example, suppose a first-derivative matrix obeys the discrete summation by parts identity $HD+D^{T}H=e_N{e}_N^T-e_0{e}_0^T$, with positive norm matrix $H$. The semidiscrete advection equation $U_t=-cDU$ then givesWith homogeneous inflow $U_0=0$ and $c>0$, only a nonpositive outflow term remains; prescribed inflow contributes a controlled forcing term. This is the inflow advection energy estimate from summation by parts, and it tests the boundary as part of the operator. For a dissipative semidiscrete matrix $A$, backward Euler also preserves the estimate, because its update givesThis distinguishes spatial energy stability from stability of an arbitrary chosen time integrator.

Finally, nonhomogeneous boundaries can be homogenized by subtracting a suitable lifting, leaving a forced equation whose solution needs a bound in terms of both initial and boundary data. Variable coefficients and nonuniform grids remove translation invariance, so freezing coefficients gives only local diagnostic information. A full energy method, a mesh-uniform operator estimate, or a dedicated boundary-symbol analysis is then needed. Fourier symbols give complete stability tests when an orthogonal mode decomposition diagonalizes the full operator; otherwise the boundaries, matrix powers and norm estimates must be checked as well.

For fixed real $\kappa$, consider initial data $u_0\in L^2(0,1)$ and a solution continuous into $L^2$ at time zero, with homogeneous Dirichlet boundary conditions for positive times. Existence, uniqueness and continuous dependence in this norm give Hadamard well-posedness.

To construct a solution, put $v(x,t)=e^{\kappa x}u(x,t)$. Substitution givesLet $b_j=2{\int }_0^1{e}^{\kappa x}{u}_0(x)\sin (j\pi x)dx$. The Fourier sine series solution isMultiplication by $e^{\pm \kappa x}$ is bounded on $L^2(0,1)$. Completeness of the sine basis gives convergence to $u_0$ in $L^2$ at time zero, while the exponential decay gives smoothness for positive times and the required equation and boundary values. For a classical solution continuous up to the endpoints at time zero, the usual initial-boundary compatibility conditions are additionally required.

For the energy method, integration by parts givesBoth boundary terms vanish. The same identity applies to the difference of two solutions, proving uniqueness and continuous dependence. The Poincaré inequality further giveswith the identical estimate for differences of initial data. This Dirichlet convection-diffusion contraction holds for every real $\kappa$, and its energy constant is independent of the drift strength.

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.

A stability analysis asks whether perturbations in an evolution remain bounded on a fixed physical time interval by a constant independent of the mesh. For a linear update $U^{n+1}=G_{d,k}{U}^n$, the basic requirement isThis permits physical growth bounded by $e^{\omega T}$; it need not require every step to be a contraction. Consistency of a numerical method measures how well its stencil reproduces the differential equation, whereas stability of a numerical method controls the accumulated defects. For a well-posed linear problem, the Lax equivalence theorem connects stability and convergence for a consistent approximation in the specified norm.

On the whole line or a periodic grid, constant-coefficient stencils commute with translations. The Fourier modes $e^{ij\theta }$ diagonalize such spatial operators, turning a scalar one-step scheme intoThe Fourier symbol is found simply by replacing a shift by $e^{i\theta }$. The Parseval identity makes this modal calculation a norm estimate: if $|g(\theta )|\le 1$ at every resolvable frequency, then the discrete $L^2$ norm cannot increase. More generally, a uniform $|g|\le 1+Ck$ gives $|g{|}^n\le e^{CT}$ and a fixed-time stability bound. All frequencies matter, including those near the grid scale; a small-wavenumber expansion establishes consistency but cannot test the entire stability range.

For the heat equation, put $r=k/d^2$ and $s={\sin }^2(\theta /2)$. The Forward Euler diffusion scheme hasThe condition $|g_{\mathrm{FE}}|\le 1$ for every $s\in [0,1]$ is exactly $0\le r\le 1/2$. This illustrates a parabolic time-step restriction $k\le d^2/2$. The Backward Euler diffusion scheme instead haswhich is contractive for every $r\ge 0$. The Crank-Nicolson diffusion scheme hasand is likewise unconditionally $L^2$ stable. However, its high-frequency amplification approaches $-1$ for large $r$, so it need not damp rapidly decaying modes effectively and may produce alternating transients from nonsmooth data. An A-stable time integrator can stabilize a diffusion semidiscretization with left-half-plane spectrum, but damping and accuracy remain separate questions.

For transport $u_t+a{u}_x=0$, define the hyperbolic Courant number $\nu =ak/d$. Forward Euler with a centered spatial difference gives $g=1-i\nu \sin \theta$ and $|g{|}^2=1+{\nu }^2{\sin }^2\theta$. It is unstable under a fixed nonzero CFL ratio. For $a>0$, the upwind finite difference scheme hasThus $0\le \nu \le 1$ is the stable range. Upwinding for negative $a$ reverses the spatial difference and gives $|\nu |\le 1$. The Lax-Wendroff advection scheme gives a second-order example:It is stable for $|\nu |\le 1$, but that $L^2$ result does not imply monotonicity or exclude oscillations at shocks. The Courant–Friedrichs–Lewy condition has a domain-of-dependence interpretation for explicit hyperbolic schemes; the Fourier calculation supplies the sharper algebraic stability criterion for the particular stencil.

For multilevel methods, each frequency gives an amplification polynomial and a companion matrix. Every amplification root must lie in the closed unit disk, with unit-modulus roots simple. Uniformity across frequencies and meshes is essential: one also needs control of the associated matrix powers, for example through a uniformly bounded diagonalization or the uniform power bound from separated amplification roots. A repeated unit root generates a Jordan block and polynomial growth in the number of steps. The same issue occurs for systems: checking only eigenvalue moduli of a nonnormal Fourier amplification matrix is insufficient. For instancehas unit eigenvalues but no fixed-time mesh-uniform bound as $n=T/k\to \infty$. For a hyperbolic system with a constant coefficient matrix admitting a uniformly bounded characteristic diagonalization, scalar modal estimates for each characteristic speed do give a system estimate. A positive energy symmetrizer provides another route. Frozen-coefficient Fourier tests for variable coefficients are useful local diagnostics, but a global estimate must also account for coefficient variation and boundary terms.

Boundary conditions change the allowable modes and can change the conclusion. A periodic calculation is exact only for a periodic closure. On a finite interval with zero Dirichlet boundary conditions, the centered diffusion matrix is diagonalized instead by the discrete sine transform. On $M$ interior points its eigenvalues are $-4{\sin }^2(j\pi /[2(M+1)])$ before scaling by $d^{-2}$. Thus the exact fixed-grid Forward Euler restriction iswhich tends to the whole-line threshold $1/2$ as the grid is refined. Backward Euler and Crank-Nicolson remain stable for all nonnegative $r$. An appropriate Neumann boundary condition uses cosine-type modes and includes a constant mode with eigenvalue zero. The constant mode must be preserved, rather than forced to decay; depending on the boundary stencil, the natural discrete inner product may have endpoint weights.

For hyperbolic transport, the continuous and discrete boundary closures must respect the direction of propagation. If $a>0$ on $[0,L]$, prescribe the left inflow value; prescribing an independent outflow value can overdetermine the problem. The energy method exposes this directly:Homogeneous inflow is dissipative, while forced inflow contributes data to the estimate. A compatible upwind boundary update has the same one-sided information flow. For diffusion,The last term vanishes for homogeneous Dirichlet or Neumann conditions. With Robin boundary conditions of dissipative sign it contributes a further nonpositive term. A non-dissipative boundary can produce genuine PDE growth, which a valid stability estimate should represent rather than incorrectly suppress.

Even a perfectly stable interior symbol cannot detect every defective boundary closure of a difference scheme. As a concrete counterexample, leave a stable explicit heat stencil on all grid points away from the left endpoint, but replace its first interior row by $U_1^{n+1}=2U_1^n$, still keeping $U_0=0$. The interior Fourier symbol is unchanged, yet $U_1^n=2^n{U}_1^0$ makes the finite-interval update unstable. The faulty boundary-adjacent row creates a mode absent from the periodic interior calculation. Thus the entire finite update matrix, or an energy estimate including its boundary rows, must be examined.

A more systematic boundary stability of a finite-difference method analysis transforms time and any tangential spatial variables, then looks for modes $U_j^n=z^n{\kappa }^j$ with $|z|>1$ and $|\kappa |<1$ on a half-line. The interior dispersion relation selects decaying spatial modes; the boundary equations must determine their coefficients without admitting a nonzero growing homogeneous mode. A uniform inverse estimate for the boundary system, expressed by the Uniform Kreiss--Lopatinskii condition, also rules out arbitrarily weak boundary control near limiting frequencies. Merely excluding one obvious unstable mode is weaker than proving a mesh-uniform estimate. Discrete summation by parts and suitable boundary penalties offer an energy-based alternative.

Finally, nonlinear scalar conservation laws can develop discontinuities, so linearized Fourier stability is only one part of the analysis. A monotone conservative scheme, such as the Engquist-Osher method under its CFL restriction, has an invariant state range and L1 contraction of a monotone conservative scheme, with a discrete entropy inequality selecting admissible shocks. These complement the Fourier and boundary estimates. For an evolution problem, the meaningful outcome is a stability bound in a specified norm, uniform over the chosen mesh family and compatible with the actual boundary and initial data.

The orthogonal projection satisfies $\Pi L=\Pi (\Pi -I)=0$, hence $d\Pi f_t/dt=0$ and $\Pi f_t=\Pi f_0=\rho (f_0)M$. Set $h_t=f_t-\Pi f_0$. Then $\Pi h_t=0$ and ${\dot{h}}_t=L{h}_t$. Crucially, applying the energy identity of part (e) to $h_t$ givesSolving this scalar equation and taking square roots provesThis argument uses (e) explicitly, rather than bypassing the requested energy method with an explicit solution formula.

Take real $\kappa$ and use the L2 norm on the spatial interval. Existence, uniqueness and continuous dependence are the three requirements of Hadamard well-posedness. An energy method supplies the decisive estimate. For a smooth solution with homogeneous Dirichlet boundary conditions, integration by parts givesThe drift contributes only a boundary term, which vanishes. The Poincaré inequality $\Vert u_x{\Vert }_2^2\ge {\pi }^2\Vert u{\Vert }_2^2$ further givesApply the same argument to the difference of two solutions to obtain uniqueness and continuous dependence on the initial data.

For existence, use the Dirichlet gauge transform for constant drift: $w=e^{\kappa x/2}u$ satisfies $w_t=w_{xx}-{\kappa }^{2}w/4$ with zero boundary values. Expanding $w_0$ in its Fourier sine series givesFor $u_0\in L^2(0,1)$ the series defines a solution continuous in $L^2$ down to $t=0$ and smooth for positive time; multiplication by the fixed bounded exponentials preserves this interpretation. Its energy estimate follows by approximation with smooth initial data. For a classical solution at the initial corners, require the usual smoothness and boundary compatibility instead. The problem is well posed in $L^2$, with a contraction estimate independent of the initial data.

A stiff differential equation can contain rapidly decaying modes as well as a slowly changing solution of interest. An explicit scheme may need a very small step solely to keep those already small fast modes from numerical growth. A-stability addresses this restriction through the Dahlquist test equation $y^{\prime }=\lambda y$, whose exact solution decays when $\mathrm{Re}\lambda <0$.

For a one-step method, write $y_{n+1}=R(z)y_n$, $z=h\lambda$. Its linear stability domain consists of the points where the update is defined and $|R(z)|\le 1$. A-stability means that this domain contains the closed left half-plane. This is a scalar linear stability property, not an order condition, an existence theorem for an implicit solve, or a general nonlinear contractivity assertion.

For a rational approximation to the exponential, consistency requires $R(z)=1+z+O(z^2)$, and order $p$ requires $R(z)-e^z=O(z^{p+1})$. Poles must be absent from the relevant half-plane. The Forward Euler method has $R=1+z$ and the disk $|1+z|\le 1$, so negative real modes require $h|\lambda |\le 2$; it is not A-stable. The Backward Euler method has $R=(1-z{)}^{-1}$, and $|1-z|\ge 1$ on the left half-plane, so it is A-stable. The trapezoidal rule and the implicit midpoint rule both have, on the scalar linear problem,They are second-order and A-stable despite being different nonlinear methods. No nonconstant polynomial approximation is A-stable, because its modulus grows without bound on the negative real axis. In particular every nontrivial explicit Runge-Kutta method has a bounded-step stability restriction.

A-stability prevents growth but need not eliminate extremely stiff modes: the trapezoidal factor tends to minus one as $z\to -\infty$. L-stability additionally requires $R(z)\to 0$ within the left half-plane. Backward Euler is L-stable; trapezoidal and implicit midpoint are not. The distinction explains persistent alternating transients in otherwise stable Crank-Nicolson calculations. Accuracy still constrains useful step sizes even for an A-stable method.

A linear multistep method is characterized by polynomials $\rho ,\sigma$, and its scalar modes satisfy $\rho (\zeta )-z\sigma (\zeta )=0$. There is generally no single scalar amplification factor: every root must lie in the closed unit disk and unit roots must be simple. At $z=0$, this is the zero-stability root condition. Consistency plus zero-stability gives convergence by the Dahlquist equivalence theorem, with suitable starting values. A-stability demands the amplification root condition throughout the left half-plane, including the zero-step condition. The Second Dahlquist barrier says that an irreducible A-stable linear multistep method has order at most two, and an explicit linear multistep method cannot be A-stable.

Examples include backward Euler, trapezoidal, and the second-order backward differentiation formulaIts zero-step roots are $1,1/3$. Its unit-circle boundary quotient has real part $(1-\cos \theta {)}^2\ge 0$, and the leading coefficient is nonzero on the left half-plane, giving A-stability by the same continuation argument as in Question 2. Conversely the third-order member of that question fails A-stability. A formal high order without zero-stability, or after silently cancelling a problematic unit factor, does not evade the barrier.

For an implicit Runge-Kutta method, the stages on the test equation satisfy $Y=e{y}_n+zAY$. Whenever these stages are solvable, its stability function isThis rational function is used to test scalar A-stability; the Butcher order conditions separately establish nonlinear order. The Lobatto IIIA method in Question 3 is an A-stable fourth-order example with $R$ the diagonal degree-two Padé approximant. More generally the $s$-stage Gauss--Legendre Runge-Kutta method has order $2s$ and is A-stable, with the diagonal Padé approximation to $e^z$; its stiff-limit factor is nonzero. The $s$-stage Radau IIA method has order $2s-1$ and is L-stable. Thus Runge-Kutta stages permit arbitrarily high A-stable order, unlike irreducible linear multistep formulas.

For a normal matrix, scalar amplification bounds transfer through unitary diagonalization. For a non-normal matrix, eigenvalue information alone does not assert Euclidean contraction: transient growth and eigenvector conditioning can matter. Direct energy method arguments, such as the dissipative Cayley-transform proof in Question 1, use the symmetric part and are more informative in that setting.

For nonlinear dissipative vector fields, the stronger B-stability property asks that distances between two numerical solutions not increase. A standard sufficient condition for Runge-Kutta methods is algebraic stability: $b_i\ge 0$ andis positive semidefinite. It follows from the Runge-Kutta contractivity identity when the implicit stages exist. The three-stage Lobatto IIIA method is A-stable but has the first diagonal entry of that matrix equal to $-1/36$, so scalar A-stability should not be confused with that sufficient nonlinear condition. Stage solvability of an implicit Runge-Kutta method remains a separate issue and computational cost of the implicit equations must be considered. A-stability removes a scalar decay-mode stability restriction; order, stiff damping, nonlinear stability and solvability remain distinct properties.

For the long-wave convection equation with broken Boussinesq symmetry, use a sufficiently smooth real temperature field. To make the spatial average and integrations meaningful, take a periodic pattern, or an existing long-interval average with bounded derivatives and vanishing averaged endpoint fluxes. Boundedness of the temperature alone does not guarantee all those averaging properties. Multiply the evolution equation by $\Theta$ and average. Integration by parts givesand $⟨\Theta ({\Theta }_x^j{)}_x⟩=-⟨{\Theta }_x^{j+1}⟩$. Thus the energy method yieldsThe energy square completion for long-wave convection starts fromPut $v={\Theta }_x$. The energy identity becomesSince $sv-v^2=s^2/4-(v-s/2{)}^2\le s^2/4$ pointwise,Therefore $\mu <2-s^2/4$ excludes growth of the mean-square temperature, for arbitrary amplitude within this smooth averaging class. This is a nonlinear energy-stability criterion, not a proof of pointwise monotonicity at each position. A spatially constant component instead decays through the $-\Theta$ term. The criterion is sufficient; it need not coincide with the linear instability threshold.

Use the Dirichlet Laplacian eigenvalues and eigenfunctions on $(-1,1)$:An expansion $u={\sum }_j{q}_j(t)e_j$ reduces the wave equation to independent ordinary differential equationsWhen $\alpha <{\lambda }_1$, put ${\omega }_j=\sqrt{{\lambda }_j-\alpha }$; each coefficient isFor finite-energy data $\varphi \in H_0^1(-1,1)$ and $\psi \in L^2(-1,1)$, a uniform spatial bound follows from the energy method, rather than from unjustified absolute summation of the Fourier series. The conserved quantity isThe Poincaré inequality gives $\Vert u{\Vert }_2^2\le {\lambda }_1^{-1}\Vert u_x{\Vert }_2^2$. Hence $2E\ge c_{\alpha }\Vert u_x{\Vert }_2^2$ with $c_{\alpha }=1-\alpha /{\lambda }_1>0$ for $0\le \alpha <{\lambda }_1$, and with $c_{\alpha }=1$ for $\alpha <0$. Since the Dirichlet boundary condition gives $u(x,t)={\int }_{-1}^x{u}_x(s,t)ds$, the Cauchy-Schwarz inequality yieldsAt $\alpha ={\lambda }_1$, the first coefficient is $q_1(t)={\varphi }_1+t{\psi }_1$, which is unbounded for some admissible data. For $\alpha >{\lambda }_1$, that coefficient has an exponentially growing component for some admissible data. Thus all-time boundedness of a wave equation with a reaction term requiresAt equality, boundedness for a particular data set requires ${\psi }_1=0$; the remaining modes are bounded by their spectral gap. Above the threshold, every unstable mode must have its growing component canceled, and any zero-frequency mode must have zero initial velocity. There is no condition on $\alpha$ alone for arbitrary specially chosen data.

The printed reference to a limit needs qualification. Bounded oscillations generally have no limit as $t\to \infty$. If actual existence of that limit for every admissible initial datum is required, no real $\alpha$ works: for every $\alpha$, choose $j$ with ${\lambda }_j>\alpha$ and nonzero pure oscillatory data in that eigenfunction. The boxed inequality answers the intended long-time boundedness question.

For energy stability for variable-coefficient reaction diffusion, let $h_x=1/(M+1)$, let $D_h$ be the Dirichlet discrete Laplacian, and set $V_h=\mathrm{diag}(a_m)$. Use the mesh-weighted inner product $(v,w{)}_h=h_x{\sum }_{m=1}^M{v}_m{w}_m$ with the corresponding discrete L2 norm. For zero boundary values, summation by parts givesFor a solution, or a difference of two solutions, the energy method givesThe Gronwall inequality therefore provesThe constant is independent of $M$, so this is stability of a numerical method on every fixed finite time interval. The bound allows physical growth if $a_{+}>0$; stability here does not mean uniform boundedness as $t\to \infty$.

There is also a uniform maximum-norm bound. At a positive spatial maximum the second difference is nonpositive. Applying this observation to $e^{-a_{+}t}v$ and to its negative gives $\Vert v(t){\Vert }_{\infty }\le e^{a_{+}t}\Vert v(0){\Vert }_{\infty }$. With a perturbing source $r(t)$, the Duhamel principle gives the corresponding initial-data bound plus ${\int }_0^t{e}^{a_{+}(t-s)}\Vert r(s)\Vert ds$ in either norm. Thus the estimate controls accumulated residuals as well as initial perturbations.

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.

The Dirichlet Laplacian eigenfunctions and positive Dirichlet Laplacian eigenvalues on the interval of length two areA Sturm-Liouville eigenfunction expansion gives $u={\sum }_j{q}_j(t){\varphi }_j$, with $q_j^{\prime \prime }+({\lambda }_j-\alpha )q_j=0$. Thus all normal modes oscillate precisely when $\alpha <{\lambda }_1$.

To justify a spatially uniform bound for arbitrary appropriate data, assume the usual wave equation energy class $u(0)\in H_0^1(-1,1)$ and $u_t(0)\in L^2(-1,1)$, or smoother compatible data. The energy method conservesThe sharp Poincaré inequality is $\Vert u{\Vert }_{L^2}^2\le {\lambda }_1^{-1}\Vert u_x{\Vert }_{L^2}^2$. For $0\le \alpha <{\lambda }_1$, this makes $E$ coercive with constant $1-\alpha /{\lambda }_1$; for $\alpha <0$, coercivity is immediate. The one-dimensional Sobolev representative satisfies $|u(x,t)|\le \sqrt{2}\Vert u_x(t){\Vert }_{L^2}$ by the fundamental theorem of calculus and the Cauchy-Schwarz inequality. Hence the conserved energy bounds $u$ uniformly in both variables.

At $\alpha ={\lambda }_1$, the smooth solution $u=t{\varphi }_1$ is unbounded. At $\alpha >{\lambda }_1$, $u=e^{\sqrt{\alpha -{\lambda }_1}t}{\varphi }_1$ is unbounded. ThereforeThe bound depends on the initial data; it is not a single bound for all arbitrarily rescaled solutions. The energy-class hypothesis supplies pointwise meaning and excludes undefined rough-data interpretations.

The spatial shift in the PDF also requires a boundary value beyond the supplied right endpoint: if $m$ is the last interior index, the newest $m+2$ value is outside the interval. Thus the printed Dirichlet method has no specified boundary closure, and no unique stability range can be assigned to it as stated.

For completeness, the literal interior formula has an exact von Neumann stability analysis on an infinite or periodic grid. Substitution of a Fourier mode $U_m^n={\xi }^n{e}^{im\theta }$ givesFor the root of a polynomial tending to one as $\theta \to 0$, expansion givesConsequently $\mu <3$ is unstable: sufficiently small nonzero frequencies have a root of a polynomial of modulus greater than one, and arbitrarily fine periodic grids contain such frequencies.