Analytic, numeric and experimental triangulation

Consider a one-dimensional ring of circumference two pi metres with a small temperature perturbation proportional to sine of x divided by one metre. Thermal diffusivity is constant, there is no advection or source, and periodic boundary conditions are imposed. The continuum heat equation keeps the spatial shape fixed while its amplitude decays exponentially. This is a benchmark with a declared initial condition, material law and boundary, rather than an arbitrary smooth curve that resembles cooling.

A spatial grid replaces the second derivative by neighboring second differences. Applying that operator to the sampled Fourier mode gives its discrete eigenvalue exactly. The continuum wavenumber squared is replaced by a sine-squared expression, which approaches it as grid spacing shrinks. To isolate that error, first imagine integrating the resulting ordinary differential equation exactly in time. A separate time integrator introduces another approximation and potentially another instability.

The discrete decay rate is smaller than the continuum rate for a nonzero resolved mode, so a coarse spatial grid cools that mode too slowly. Refinement should reduce this discrepancy with the predicted second-order trend. This verifies a numerical approximation to the specified equation. It does not show that the equation correctly describes a real ring, whose contact losses, sensor averaging or temperature-dependent material properties might differ from the benchmark assumptions.

An experimental amplitude ratio supplies a rate through its logarithm. Inferring diffusivity from that rate assumes the wavenumber, elapsed time and measurement response are known. A constant unknown gain cancels from a ratio, whereas an additive sensor offset does not. If a coarse simulation is fitted by increasing diffusivity, the fit can hide its spatial error inside a biased material parameter. The appropriate comparison therefore includes both grid refinement and independent information about the physical measurement process.

A good comparison records what is common among the three routes. The analytic and grid calculations share the heat equation, so their agreement cannot independently validate Fourier’s constitutive law. Two sensors may share one calibration error, and two codes may share one mistaken boundary condition. Independence is a physical and mathematical property of the evidence, not a count of plots. Design an additional comparison around a failure mode that the existing routes share.

For example, compare several spatial wavelengths to test the predicted wavenumber-squared rate, or reverse an imposed flow to isolate advection. Record whether a reported disagreement exceeds thermometry and discretization uncertainty before changing the transport model. A persistent discrepancy may motivate a new physical term, but that term is initially a proposal. Its derivation, parameter calibration and prediction for an unused observable are separate stages. The finite examples below make the numerical part of this reasoning directly checkable.