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Write $h=\Delta x$ and set $u_0=u_{M+1}=0$. In the discrete L2 norm $\Vert u{\Vert }_h^2=h{\sum }_{m=1}^M|u_m{|}^2$, the energy method givesTo obtain this identity, shift the indices in the centered second-difference sum. The centered first-difference matrix is skew-symmetric, so its contribution has zero real part even for complex grid values. The homogeneous Dirichlet boundary conditions remove the endpoint terms.

Therefore the semidiscretization is stable for every real $\alpha$ and every positive $h$, with the mesh-independent bound $\Vert u(t){\Vert }_h\le \Vert u(0){\Vert }_h$. No restriction on $|\alpha |h$ is needed for this energy estimate.

Let $L_h=D_{xx}+\alpha D_x$, where the centered second-difference matrix $D_{xx}$ is real symmetric negative definite under homogeneous Dirichlet boundary conditions, and the centered first-difference matrix $D_x$ is real skew-symmetric. HenceThe semidiscrete energy method gives $\Vert \mathbf{u}(t){\Vert }_2\le \Vert \mathbf{u}(0){\Vert }_2$ for every real $\alpha$, so the scheme is stable.

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 $h\to 0$. 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 regionA-stability means that $\mathcal{S}$ contains the entire closed left half-plane, matching every decaying scalar linear problem. L-stability additionally requires $R(z)\to 0$ 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 $|1+z|\le 1$.

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 $BA+A^{T}B-b{b}^T$—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.

For a numerical evolution, numerical stability means that perturbations remain controlled over a fixed finite interval by a bound independent of sufficiently small discretization parameters. With perturbations ${\delta }_n$ to starting data and added step defects $d_n$, a typical estimate iswhere $C_T$ is independent of $h$. For a linear homogeneous recurrence, a uniform bound on powers of its update matrix supplies this control. von Neumann stability analysis and the energy method provide complementary tools.

Finite-interval numerical stability differs from absolute stability at a fixed step size. Applying the Euler method to $y^{\prime }=-2y$ multiplies a perturbation by $1-2h$ each step. At $h=0.2$ this is $0.6$, so perturbations shrink; at $h=1.2$ it is $-1.4$, so they alternate and grow. Nevertheless the Euler method converges as $h\to 0$ for this problem: failure at a coarse step does not contradict the convergence theorem. Absolute stability asks whether a selected step preserves boundedness of the Dahlquist test equation over arbitrarily many steps. MIT's analysis uses strict decay, $|1+h\lambda |<1$, rather than the boundary-inclusive boundedness convention.^[ref]

Let $\mu >0$ and let $u$ be a real smooth solution on a bounded regular domain with zero Dirichlet boundary conditions. Green's first identity givesThe two sides have opposite signs unless both vanish, so $u=0$. This energy method also proves uniqueness for prescribed boundary data by applying it to the difference of two solutions. For complex solutions use the complex conjugate in the integral. This argument does not apply with the same sign to the Helmholtz equation, which can have nonzero zero-boundary eigenfunctions.