Finite-difference wave propagation and CFL

For displacement u and positive wave speed c, replace both second derivatives by centered differences. The resulting update uses two previous time levels, so initial displacement alone cannot start it. A Taylor expansion using the prescribed initial velocity and the wave equation produces a consistent first time level. The discrete spatial Laplacian in this initialization must respect the same boundary convention as later steps. Treating a missing neighbor as zero when the model is periodic changes the problem.

A periodic grid wraps the last point back to the first and contains no distinct duplicated endpoint. Fixed Dirichlet ends instead prescribe displacement at the boundary, while a Neumann end requires a consistent derivative stencil. Ghost cells are one implementation device, not new physical particles. Even if an interior formula is second order, a boundary or startup error can control the observed accuracy. Declare the entire initial-boundary-value problem rather than attach a boundary rule after inspecting the animation.

On an infinite or periodic uniform grid, translation invariance allows a single Fourier grid mode to be examined independently. Substituting its spatial oscillation and temporal amplification factor produces a quadratic equation for that factor. For the central one-dimensional wave update, the usual Courant limit is one. Above it, a sufficiently short resolved wave has an exponentially growing branch, so roundoff or a small startup perturbation can seed an instability even when the intended initial pulse looks smooth.

The precise condition belongs to this equation and discretization. Higher dimensions, nonuniform grids, variable coefficients and different time methods alter it. At the endpoint one should also initialize compatible displacement and velocity carefully; repeated-root modes need the proper stability interpretation. Using a strict margin below the limit is a practical choice, not a replacement for analysis. A domain-of-dependence argument gives a necessary restriction for many explicit methods, but by itself does not prove every scheme stable.

A stable mode can travel at the wrong numerical speed. Its frequency obeys a sine relation instead of the continuum linear relation, so short wavelengths acquire different phase errors. For smooth solutions the centered derivative consistency error is second order in time and space. Stability plus consistency implies convergence only under the relevant well-posed linear problem and norm assumptions, including compatible boundaries and startup. These concepts should not be collapsed into the statement that a small step makes every calculation correct.

Refining space while leaving time fixed can expose temporal error, whereas reducing time alone cannot resolve a spatially under-sampled wave. Compare at the same physical time and examine a norm or phase observable appropriate to the problem. The waves lab adds two exact same-frequency sine waves. It can demonstrate cancellation and superposition, but has no spatial mesh, Courant number or numerical dispersion; it cannot display this difference scheme becoming unstable.