Finite differences: consistency and stability

On a uniform grid x_j=jh and times t_n=nk, approximate the heat equation by a forward time difference and a centered second spatial difference. The update is U_j^(n+1)=U_j^n+r(U_(j+1)^n−2U_j^n+U_(j−1)^n), where r=κk/h². Insert a sufficiently smooth exact solution into the difference equation, divide by k in the residual convention, and expand in Taylor series. The residual is O(k+h²) on a bounded time interval if the needed time and fourth spatial derivatives stay uniformly bounded.

That statement concerns the scheme's agreement with the PDE on smooth inputs, not the computed solution's global error. It also depends on implementing initial and boundary data consistently. A second-order interior stencil can still have poor overall accuracy if a boundary approximation is first order or imposes the wrong condition. The time and spatial step sizes are not independent when stability restricts their ratio. Refining only h with k fixed can make the method less stable even though the centered spatial derivative becomes more accurate as a local formula.

For a periodic grid, insert a Fourier error mode exp(ijθ) into the linear update. Its amplification factor is G(θ)=1−4r sin²(θ/2). Discrete L2 stability over repeated time steps requires |G(θ)|≤1 for all resolvable frequencies, giving 0≤r≤1/2. At r=1/2 the highest alternating mode has G=−1, so it changes sign without decaying. For r>1/2 that mode has magnitude greater than one and grows. A stable discretization need not reproduce every physical decay rate exactly, especially near the resolution limit.

There is also a direct maximum-norm argument when 0≤r≤1/2: each new grid value is a convex combination of its old value and its two neighbors. With compatible boundary handling, it cannot create a larger absolute maximum than the data already supply. Outside that range the central coefficient is negative and this argument fails. Fourier analysis here uses periodicity; a bounded interval with other boundary stencils requires the stability analysis of the complete discrete operator, not a claim that the interior symbol alone proves everything.

The Lax–Richtmyer equivalence theorem concerns consistent linear approximations of a well-posed linear initial-value problem in an appropriate normed-space framework. With its approximation and time-step hypotheses, stability is equivalent to convergence on fixed finite time intervals. The proof tracks an error recurrence: each step introduces a small truncation defect, and a uniform bound on the discrete evolution prevents accumulated defects from exploding. This is not an unconditional guarantee for every nonlinear scheme, arbitrary discontinuous solution, or changing boundary treatment. Nonlinear conservation laws need additional structure such as entropy consistency and compactness.

The optimization lab provides an inspectable scalar amplification analogy. Its quadratic gradient iteration multiplies x by 1−η each step. Choose η below two, then above two, to compare shrinking and growing iterates; reset before making a comparison. This is not the heat stencil, and η is not automatically the heat ratio r. For one particular Fourier mode, the analogy uses η=4r sin²(θ/2), which changes with frequency. The lab illustrates repeated multiplication; the stencil analysis supplies the full range of modes and its PDE-specific stability restriction.