= Solution
Stability asks whether perturbations already present in data, arithmetic, or an earlier numerical step remain controlled under subsequent evolution. It complements <consistency of a numerical method>: consistency says that an exact smooth solution nearly satisfies one numerical step, while stability prevents the accumulated local defects from being amplified without bound. For a well-posed differential equation these two properties are what make convergence possible.
For a <linear multistep method>, zero-stability concerns the limit $h\to0$. The roots of its first characteristic polynomial must lie in the closed unit disk, and every root on the unit circle must be simple. The <Dahlquist equivalence theorem> states that a consistent linear multistep method converges precisely when it is zero-stable. The parasitic root $5/11$ in Question 1 decays, whereas a repeated root at $1$ would turn small defects into secular growth.
Absolute stability instead fixes $z=h\lambda$ for the <Dahlquist test equation>. A one-step method has amplification factor $R(z)$ and absolute-stability region
$$
\mathcal S=\{z:|R(z)|\leq1\}.
$$
A-stability means that $\mathcal S$ contains the entire closed left half-plane, matching every decaying scalar linear problem. <L-stability> additionally requires $R(z)\to0$ as $z\to-\infty$, which suppresses unresolved fast transients in a <stiff differential equation>. Backward Euler is L-stable; the trapezoidal rule is A-stable but approaches $-1$ and can retain stiff oscillations; forward Euler is stable only in the disk $|1+z|\leq1$.
For nonlinear dissipative systems, scalar absolute stability can be insufficient. B-stability controls distances between numerical solutions of contractive differential equations. For Runge–Kutta methods, algebraic stability—nonnegative weights and positive semidefiniteness of $BA+A^TB-bb^T$—is a useful sufficient condition for B-stability. Question 2 shows how a single negative diagonal entry disproves it.
After spatial discretization of a partial differential equation, stability can be studied through the semidiscrete matrix. If its Hermitian part is nonpositive, the <energy method> proves contractivity without diagonalizing it. A skew-Hermitian generator instead conserves norm, as in the discrete Schrodinger problem of Question 3. Eigenvalues alone can be misleading for a <non-normal matrix>: transient growth may be large even when every eigenvalue lies in the left half-plane, so matrix norms, logarithmic norms, resolvent bounds, or a <pseudospectrum> may be needed.
For constant-coefficient Cauchy problems, <Von Neumann stability analysis> inserts Fourier modes and bounds their amplification factors. Question 4 gives a typical result: centered diffusion contributes a negative real symbol and centered advection an imaginary symbol. Semidiscrete evolution is stable, yet forward Euler imposes both the parabolic restriction $\Delta t=O(\Delta x^2)$ and an advection-diffusion restriction. This illustrates a <Courant–Friedrichs–Lewy condition>: a stable spatial approximation need not remain stable under an arbitrary time stepper.
For a well-posed linear initial-value problem and a consistent finite-difference approximation, the <Lax equivalence theorem> identifies stability with convergence. Its hypotheses matter: it does not by itself cover nonlinear equations, inconsistent boundary closures, changing norms, or non-smooth solutions. Boundaries also defeat a naive whole-line Fourier argument; an energy estimate, normal-mode boundary analysis, or a discrete semigroup bound must include the boundary treatment.
Practical stability analysis therefore starts from the structure of the differential equation. Conservation laws favor unitary or symplectic methods, diffusion favors A- or L-stable implicit methods, monotone transport may require strong-stability-preserving time stepping and upwind fluxes, and stiff splitting requires attention to both the factors and their commutators. Stability does not guarantee accuracy: a heavily damped method may be stable while erasing the solution, and a stable computation with $h\lambda$ near the edge of its stability region may have an unacceptable phase error.
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