= Solution
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> means
$$
\boxed{\|C_{h,k}^{\,n}\|\leq C_T\quad\text{whenever }nk\leq T},
$$
with $C_T$ 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> $U_j^n=G(\theta)^ne^{ij\theta}$. 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, $|G(\theta)|\leq1$ proves contractivity, while $|G(\theta)|\leq1+\omega k$ 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 $\mu=k/h^2$,
$$
G(\theta)=1-4\mu\sin^2(\theta/2).
$$
Thus $|G|\leq1$ for all frequencies exactly when
$$
\boxed{0\leq\mu\leq\frac12}.
$$
Sufficiency 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 $k\leq h^2/2$. For the <Backward Euler diffusion scheme>,
$$
G(\theta)=\frac1{1+4\mu\sin^2(\theta/2)},
$$
so every $\mu\geq0$ is stable. The <Crank-Nicolson method> has
$$
G(\theta)=\frac{1-2\mu\sin^2(\theta/2)}{1+2\mu\sin^2(\theta/2)},
$$
again of <modulus> at most one for all $\mu\geq0$, 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 $U_j^{n+1}=(1-\nu)U_j^n+\nu U_{j-1}^n$, with
$$
|G(\theta)|^2=1-2\nu(1-\nu)(1-\cos\theta).
$$
It is contractive when $0\leq\nu\leq1$. 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 $\log|G|\leq a^2k^2/(2h^2)$.
The <leapfrog advection scheme> illustrates a multilevel subtlety. Its <amplification polynomial of a multilevel finite difference scheme> is
$$
\xi^2+2i\nu\sin\theta\,\xi-1=0.
$$
For $|\nu|<1$, 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 $\|U\|_h^2=h\sum_j|U_j|^2$. The discrete <summation by parts> identity is
$$
\langle U,L_hU\rangle_h=-\|D_+U\|_h^2.
$$
Take the <inner product> of $U^{n+1}-U^n=kL_hU^{n+1}$ with $2U^{n+1}$. The elementary identity $2\operatorname{Re}\langle x-y,x\rangle=\|x\|^2-\|y\|^2+\|x-y\|^2$ gives
$$
\|U^{n+1}\|_h^2-\|U^n\|_h^2
+\|U^{n+1}-U^n\|_h^2+2k\|D_+U^{n+1}\|_h^2=0.
$$
This proves unconditional contractivity including the actual boundary treatment. More generally, a <dissipative operator> $L_h$ gives $(I-kL_h)^{-1}$ of <operator norm> at most one: with $V=(I-kL_h)U$, dissipativity implies $\operatorname{Re}\langle V,U\rangle\geq\|U\|^2$, and the <Cauchy-Schwarz inequality> gives $\|U\|\leq\|V\|$. 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 $U'=L_hU$ by $R(kL_h)$. 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,
$$
C_{h,k}=\begin{pmatrix}1-k&k/h\\0&1-k\end{pmatrix}
$$
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 $Te^{-T}/h$. 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 $k\leq k_{\mathrm{FE}}$, the second-order <strong stability preserving Runge-Kutta method>
$$
Y=E_k(U),\qquad
U_{\mathrm{new}}=\frac12U+\frac12E_k(Y)
$$
is also nonexpansive under the same restriction. Indeed, for two inputs, the <triangle inequality> gives $\|U_{\mathrm{new}}-\widetilde U_{\mathrm{new}}\|\leq\frac12\|U-\widetilde U\|+\frac12\|E_k(Y)-E_k(\widetilde Y)\|\leq\|U-\widetilde U\|$. Its <Taylor expansion> is $U+kF(U)+k^2F'(U)F(U)/2+O(k^3)$, 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 principle>
$$
e^n=C_{h,k}^ne^0+k\sum_{j=0}^{n-1}C_{h,k}^{n-1-j}\tau^j,
\qquad
\|e^n\|\leq C_T\bigl(\|e^0\|+T\max_j\|\tau^j\|\bigr).
$$
Thus 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. \b[The decisive checks are a mesh-uniform evolution bound, a valid boundary closure, and consistency with the actual PDE.]
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