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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 $\hat{U}(\theta )={\sum }_j{U}_j{e}^{-ij\cdot \theta }$, its symbol is $G(\theta )={\sum }_{\ell }{a}_{\ell }{e}^{i\ell \cdot \theta }$ and ${\hat{U}}^{,n}=G^n{\hat{U}}^0$. Parseval identity givesThe 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 $|G|\le 1$ at every frequency. For mesh-uniform stability on $nk\le T$, a sufficient condition is instead $\sup |G|\le 1+Ck$, because $(1+Ck{)}^n\le e^{CT}$. Conversely, if the powers at $n=\lfloor T/k\rfloor$ are bounded by a common $M_T\ge 1$, then $\sup |G|\le M_T^{1/n}=1+O(k)$ as $k\to 0$. Thus finite-time stability may allow mild modal growth; demanding strict contraction is stronger than necessary.

Stability connects numerical consistency to numerical convergence. Let $R_{h}u(t_n)$ be the exact solution restricted by a suitable projection and let its one-step defect be ${\tau }_n=R_{h}u(t_{n+1})-S_{h,k}{R}_{h}u(t_n)$. If $\Vert {\tau }_n\Vert \le Ck(h^p+k^q)$ and $\Vert S^n\Vert \le M_T$, the error recurrence givesThis 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 $R_h{J}_h=I$, and suppose the grid and reconstructed norms are uniformly equivalent. numerical convergence for every datum, uniformly on the fixed time interval, makes the operators $J_h{S}_{h,k}^n{R}_h$ pointwise bounded. The Uniform boundedness principle makes their operator norms uniformly bounded. Apply this bound to $J_{h}V$, use $R_h{J}_{h}V=V$ 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,Its minimum is $1-4\mu$, so $0\le \mu \le 1/2$ 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 ${\sum }_a{\mu }_a\le 1/2$, or $\mu \le 1/(2d)$ 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, $\log |G|\le a^2{k}^2/(2h^2)=O(k)$, allowing finite-time stability. This illustrates why the specified refinement path matters. For $a>0$, the upwind update instead hasand is contractive exactly for $0\le \nu \le 1$. 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 $G=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ has both eigenvalues one but $G^n=\left(\begin{array}{cc}1& n\\ 0& 1\end{array}\right)$ is unbounded as the number of steps increases. If $G=X\Lambda X^{-1}$ with uniformly bounded condition numbers, then $\Vert G^n\Vert \le \Vert X\Vert \Vert X^{-1}\Vert \max |\lambda {|}^n$, 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.