= Solution
For constant-coefficient evolution on an infinite or periodic grid, spatial <Fourier stability analysis> reduces a multidimensional recurrence to independent modal problems. This gives exact L2 <stability> results when the coefficients and geometry admit that reduction; it also exposes high-frequency restrictions which a truncation-error calculation alone misses.
Consider a scalar one-step update $(S_{h,k}U)_j=\sum_\ell a_\ell U_{j+\ell}$ on $\mathbb Z^d$. With $\widehat U(\theta)=\sum_jU_je^{-ij\cdot\theta}$, its symbol is $G(\theta)=\sum_\ell a_\ell e^{i\ell\cdot\theta}$ and $\widehat U^{,n}=G^n\widehat U^0$. <Parseval identity> gives
$$
\|U\|_h^2=\frac{h^d}{(2\pi)^d}\int_{[-\pi,\pi]^d}|\widehat U(\theta)|^2d\theta,
\qquad \boxed{\|S_{h,k}^n\|=\mathop{\mathrm{ess\,sup}}_\theta|G(\theta)|^n.}
$$
The upper bound is immediate by integration. For the reverse inequality choose Fourier data supported in a positive-measure neighborhood where $|G|$ approaches its essential supremum, then normalize them. This uses actual square-summable data, unlike an isolated infinite plane wave. On a periodic finite grid the <discrete Fourier transform> gives the same equality with a maximum over its resolved frequencies.
This proves the <scalar Fourier power criterion>. Contractivity is equivalent to $|G|\leq1$ at every frequency. For mesh-uniform <stability> on $nk\leq T$, a sufficient condition is instead $\sup|G|\leq1+Ck$, because $(1+Ck)^n\leq e^{CT}$. Conversely, if the powers at $n=\lfloor T/k\rfloor$ are bounded by a common $M_T\geq1$, then $\sup|G|\leq M_T^{1/n}=1+O(k)$ as $k\to0$. Thus finite-time <stability> may allow mild modal growth; demanding strict contraction is stronger than necessary.
<Stability> connects <numerical consistency> to <numerical convergence>. Let $R_hu(t_n)$ be the exact solution restricted by a suitable projection and let its one-step defect be $\tau_n=R_hu(t_{n+1})-S_{h,k}R_hu(t_n)$. If $\|\tau_n\|\leq Ck(h^p+k^q)$ and $\|S^n\|\leq M_T$, the error recurrence gives
$$
e_n=S^ne_0-\sum_{j=0}^{n-1}S^{n-1-j}\tau_j,
\qquad
\boxed{\|e_n\|\leq M_T\|e_0\|+CM_TT(h^p+k^q).}
$$
This proves <numerical convergence> for smooth solutions and convergent initial approximations. Density plus <stability> extends <numerical convergence> to the natural data space when stable projections and well-posed continuous evolution are available.
For completeness, the converse in the <Lax equivalence theorem> can be understood with faithful grid transfers. Let $J_h$ be a bounded reconstruction into the continuous data space and $R_h$ a bounded projection with $R_hJ_h=I$, and suppose the grid and reconstructed norms are uniformly equivalent. <numerical convergence> for every datum, uniformly on the fixed time interval, makes the operators $J_hS_{h,k}^nR_h$ pointwise bounded. The <Uniform boundedness principle> makes their <operator norms> uniformly bounded. Apply this bound to $J_hV$, use $R_hJ_hV=V$ and the norm equivalence, and obtain the uniform grid-power bound. With the <numerical consistency> estimate above, this proves the <stability>/<numerical convergence> equivalence under these standard linear assumptions.
Several examples illustrate the test. For forward time and centered space on the one-dimensional <heat equation>,
$$
G(\theta)=1-4\mu\sin^2(\theta/2),\qquad \mu=k/h^2.
$$
Its minimum is $1-4\mu$, so \b[$0\leq\mu\leq1/2$] is the exact contraction range. In $d$ dimensions with mesh widths $h_a$, the symbol is $1-4\sum_a\mu_a\sin^2(\theta_a/2)$, where $\mu_a=k/h_a^2$. Therefore \b[$\sum_a\mu_a\leq1/2$], or $\mu\leq1/(2d)$ on an equal mesh. This follows by independently maximizing every sine factor, not by applying a one-dimensional bound separately to each direction.
For advection $u_t+a u_x=0$, forward time and centered space give $G=1-i\nu\sin\theta$, $\nu=ak/h$. At fixed nonzero advective <Courant number> its modulus exceeds one, so its powers grow without a mesh-uniform bound. Under the much smaller choice $k=O(h^2)$, however, $\log|G|\leq a^2k^2/(2h^2)=O(k)$, allowing finite-time <stability>. This illustrates why the specified refinement path matters. For $a>0$, the upwind update instead has
$$
G=1-\nu+\nu e^{-i\theta},\qquad
|G|^2=1-4\nu(1-\nu)\sin^2(\theta/2),
$$
and is contractive exactly for $0\leq\nu\leq1$. Its numerical diffusion explains the stable damping as well as its first-order spatial accuracy.
For the <heat equation> with the <Crank-Nicolson method>, the symbol is $(1-2\mu\sin^2(\theta/2))/(1+2\mu\sin^2(\theta/2))$. It has modulus at most one for every nonnegative $\mu$, so there is no diffusion time-step restriction. Very stiff modes nevertheless have amplification near minus one, causing weakly damped oscillation; <stability> alone does not guarantee good stiff smoothing or small error.
For systems the symbol is a <matrix>, and the exact criterion is the <uniform power bound for matrix Fourier symbols>, not just a bound on <eigenvalue> moduli. For example $G=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ has both <eigenvalues> one but $G^n=\begin{pmatrix}1&n\\0&1\end{pmatrix}$ is unbounded as the number of steps increases. If $G=X\Lambda X^{-1}$ with uniformly bounded condition numbers, then $\|G^n\|\leq\|X\|\|X^{-1}\|\max|\lambda|^n$, so a scalar-type condition does suffice. Multilevel schemes are treated by stacking their time levels into a companion <matrix> and checking its uniform powers. Simple unit-modulus roots and interior roots are the fixed-mode root condition, but uniformity near colliding roots still needs attention.
The technique is inexpensive, identifies the dangerous wavelengths, gives sharp constant-coefficient restrictions and generalizes directly to several space variables and Fourier-diagonalizable vector problems. Its limitations are equally concrete: boundaries can add unstable modes invisible to an infinite-grid calculation; variable coefficients couple frequencies; nonlinearities invalidate the fixed multiplier; nonuniform grids need another transform or energy estimate; and L2 <stability> does not by itself imply a maximum principle or positivity. Frozen-coefficient analysis is often useful evidence, while an energy or boundary estimate is needed to turn it into a global conclusion in those settings.
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