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Let $s_{\ast }$ be the unique sonic point, $f^{\prime }(s_{\ast })=0$. For a differentiable convex function, $f^{\prime }$ is nondecreasing, negative below $s_{\ast }$ and positive above it. Split the flux into the parts propagating right and left:Thus $f(u)=f(s_{\ast })+f^{+}(u)+f^{-}(u)$; explicitly, $f^{+}(u)=f(u)-f(s_{\ast })$ for $u>s_{\ast }$ and zero below, while $f^{-}(u)=f(u)-f(s_{\ast })$ for $u<s_{\ast }$ and zero above. The Engquist-Osher flux takes the right-going contribution from the left state and the left-going contribution from the right state:It is consistent, $F(u,u)=f(u)$. For a transonic rarefaction wave with $a\le s_{\ast }\le b$ it gives $f(s_{\ast })$; reversed transonic states give $f(a)+f(b)-f(s_{\ast })$. This is sonic-point splitting of a convex Engquist-Osher flux.

On cells of width $h$ let $U_j^n$ approximate the cell average and $\lambda =\Delta t/h$. The explicit Engquist-Osher method is the finite volume methodThe shared interface flux makes it conservative: sums telescope for periodic data, and the same applies on the infinite grid when boundary contributions vanish. To prove stability, choose an interval $[m,M]$ containing the initial states and impose the CFL conditionThe partial derivatives of the update areTherefore the update is a monotone conservative scheme. Because $H(q,q,q)=q$, componentwise comparison with the constant states $m,M$ proves $m\le U_j^{n+1}\le M$. Induction gives a mesh-independent maximum bound for every time level, and ensures the same CFL speed bound remains valid.

There is also a stronger L1 contraction of a monotone conservative scheme. Let $T$ denote one whole-grid update, and let $W=\min (U,V)$ and $Z=\max (U,V)$ componentwise. Monotonicity gives $TW\le TU,TV\le TZ$. Hence $|TU-TV|\le TZ-TW$ componentwise, and conservation yieldsThis applies to periodic grids or summable perturbations with vanishing end fluxes. Apply the same result to $U$ and its one-cell translate; translation commutes with $T$, soThus the method is a total variation diminishing scheme. Under the stated CFL condition it preserves the initial range, contracts differences in discrete $L^1$, and cannot increase total variation. These bounds give a precise nonlinear stability result, including across the sonic point.

Convexity alone does not remove the time-step restriction omitted in the question. For the Inviscid Burgers equation $f(u)=u^2/2$, linearize about a positive constant $U$. The update becomes $v_j^{n+1}=(1-\lambda U)v_j^n+\lambda U{v}_{j-1}^n$. Its alternating Fourier mode has multiplier $1-2\lambda U$, whose modulus exceeds one if $\lambda U>1$. Hence unrestricted explicit time steps are unstable even for a convex flux with one sonic point. This is CFL necessity for explicit Engquist-Osher stability.

The Engquist-Osher method is an upwind finite volume method. For cell values $U_j^n$, mesh width $h$ and time step $k$, put $r=k/h$ and use the conservative updatewhere the numerical flux isConsistency of a numerical method follows from $F(u,u)=f(u)$. Positive characteristic speeds use the left state, and negative speeds use the right state. The flux difference telescopes when summed over cells, giving a discrete conservation law. The method is first order in space and time for smooth solutions; its conservation law form is also appropriate when shock waves develop.

Let $s_{\ast }$ be the unique sonic point of a scalar flux, $f^{\prime }(s_{\ast })=0$. Convexity implies $f^{\prime }<0$ to its left and $f^{\prime }>0$ to its right. Put $f_{\ast }=f(s_{\ast })$ and splitThen $f=f_{\ast }+f^{+}+f^{-}$, $(f^{+}{)}^{\prime }\ge 0$, $(f^{-}{)}^{\prime }\le 0$, andThis is the sonic-point splitting of a convex Engquist-Osher flux. If both states are above $s_{\ast }$, $F=f(a)$; if both are below it, $F=f(b)$. Across a transsonic rarefaction wave, $a\le s_{\ast }\le b$, the flux is the minimum $f_{\ast }$. In the opposite crossing it is $f(a)+f(b)-f_{\ast }$, which in general differs from the exact Godunov numerical flux. This distinction does not affect the monotonicity proof.

Here stability requires the usual Courant–Friedrichs–Lewy condition. Let the initial values lie in $[m,M]$, and chooseA differentiable convex function has a continuous derivative on this interval, so the maximum is finite. Write the three-input update as $H(a,b,c)$. Differentiating givesThus the update $T$ preserves componentwise order. Constants are fixed, since $H(c,c,c)=c$, so comparison with the constants $m,M$ proves $m\le U_j^n\le M$ for every time level. In particular the same CFL condition remains applicable. This is a mesh-independent maximum-principle stability bound.

A stronger stability statement is contraction in discrete $L^1$. Consider two solutions $U,V$ in the same invariant interval, on a periodic mesh or on the infinite lattice with summable differences. Let $Q=U\vee V$ be their componentwise maximum. Monotonicity gives $TQ\ge TU,TV$, henceThe discrete conservation law for the difference gives the equality: on a finite periodic mesh the flux sum cancels exactly, and on the infinite lattice it follows by truncation and the fact that the flux is a Lipschitz function on $[m,M]$. Interchanging $U,V$ and adding provesThis is L1 contraction of a monotone conservative scheme. Since $T$ commutes with the cell shift, applying this contraction to $U$ and its shift gives ${\sum }_j|U_{j+1}^{n+1}-U_j^{n+1}|\le {\sum }_j|U_{j+1}^n-U_j^n|$. The scheme is therefore a total variation diminishing scheme as well.

The same order argument proves a discrete entropy inequality, explaining why this stable discretization selects entropy solutions rather than nonphysical expansion shock waves. For a constant $c\in [m,M]$, setBecause $T(U\vee c)\ge TU,c$ and $T(U\wedge c)\le TU,c$, their componentwise difference is at least $|TU-c|$. Subtracting their updates givesFor constants outside $[m,M]$, the entropy inequality is an equality by the invariant bound and the conservative update. At equal states $Q_c(u,u)=\mathrm{sgn}(u-c)[f(u)-f(c)]$, the Kruzhkov entropy flux. The flux is a Lipschitz function on the invariant interval: writing $L=\max |f^{\prime }|$, the conservative update gives $h{\sum }_j|U_j^{n+1}-U_j^n|\le 2kL{\sum }_j|U_{j+1}^n-U_j^n|$. This time-translation bound, the invariant bound and the total variation estimate give local space-time compactness for bounded-variation initial conditions as the mesh is refined; the conservative update passes to the weak conservation law, and the displayed inequality passes to its Kruzhkov entropy inequality. This identifies the limit as the entropy solution.

Finally the time-step qualification is essential. For the convex Inviscid Burgers equation flux $f(u)=u^2/2$, whose unique sonic point of a scalar flux is zero, linearize about a positive constant $U$. The perturbation update becomes $v_j^{n+1}=(1-rU)v_j^n+rU{v}_{j-1}^n$. At Fourier frequency $\pi$ its multiplier is $1-2rU$, with modulus greater than one if $rU>1$. Wave packets near this phase give linear instability even for summable Cauchy perturbations. Thus the assumptions on $f$ imply the stability results above under the CFL condition, not for arbitrary $k/h$. This is CFL necessity for explicit Engquist-Osher stability.

The Engquist-Osher method is a conservative finite volume method built by separating positive and negative characteristic speeds. Take $f\in C^1$, a uniform cell width $d$, a time step $k$, and cell averages $U_j^n$. Write $\lambda =k/d$ and define the numerical fluxThe Engquist-Osher flux uses the left state for right-going transport and the right state for left-going transport. Its conservative update isIt is consistent because $F(u,u)=f(u)$. The integrals have their ordinary oriented meaning even for negative states. For the Inviscid Burgers equation, $f(u)=u^2/2$ gives the useful exampleIn a region where all characteristic speeds have one sign, the flux reduces to the corresponding upwind flux. In smooth regions, this basic piecewise-constant, forward-time version is first order in space and time; its main virtue is robust nonlinear stability across shocks.

Here is a precise stability proof. Assume the data lie in a bounded interval $J=[m,M]$ and chooseWork on a periodic grid, or on the infinite grid with summable differences; on a finite interval the inflow boundary data and boundary fluxes must be treated monotonically as well. Define the three-point update mapWriting $f_{+}^{\prime }=\max (f^{\prime },0)$ and $f_{-}^{\prime }=\min (f^{\prime },0)$, its three partial derivatives areThus it is a monotone conservative scheme. Since $H(s,s,s)=s$, comparison with the constant states provesThe interval is invariant, so the same Courant–Friedrichs–Lewy condition remains valid at every step. This is an $L^{\infty }$ bound relative to constant states, not a claim that arbitrary pairs of solutions are contractive in $L^{\infty }$.

For a stronger perturbation estimate, let $\mathcal{T}$ denote the global update and use componentwise maxima. Monotonicity gives $\mathcal{T}(U\vee V)\ge \mathcal{T}U,\mathcal{T}V$. ThereforeSum over the grid. The shared interface flux cancels telescopically, so conservation givesInterchanging $U,V$ and adding yields the L1 contraction of a monotone conservative scheme:This is a mesh-independent nonlinear stability estimate. On an infinite grid the telescoping argument is justified by summability of the differences and the bounded characteristic-speed range, or by truncation followed by a limit. For unequal prescribed boundary data, additional boundary flux terms enter the estimate.

The method is also a total variation diminishing scheme. Let $\mathcal{S}$ shift a grid sequence by one cell. Translation invariance gives $\mathcal{T}\mathcal{S}=\mathcal{S}\mathcal{T}$. Applying the preceding $L^1$ contraction to $U$ and $\mathcal{S}U$ showsHence initially finite discrete total variation cannot grow.

Finally, monotonicity supplies a discrete entropy inequality, explaining why the stability is compatible with the physically admissible entropy solution. For a constant $c\in J$, defineComparison of $H$ with the clipped triples above and below $c$ givesIndeed $H(U\vee c)\ge \max (H(U),c)$ and $H(U\wedge c)\le \min (H(U),c)$, and subtracting these two update formulas yields exactly the right side. Also $Q(u,u;c)=\mathrm{sgn}(u-c)(f(u)-f(c))$, the Kruzhkov entropy flux. Under the displayed CFL condition the method therefore has an invariant state range, $L^1$ contraction, decreasing discrete total variation and entropy dissipation. These estimates remain meaningful at discontinuities, where linearized Fourier analysis alone cannot establish the corresponding nonlinear stability.

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.