A metal rod's temperature diffuses with

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Computational Physics, Ch Rules on a lattice: diffusion, percolation and growth

Problem

A metal rod's temperature diffuses with D=2.0×10−3 m2 s−1D = 2.0\times10^{-3}\,\mathrm{m^2\,s^{-1}}, known independently from the rod's thermal properties. Two explicit-scheme runs at the same fixed diffusion number α\alpha (with Δt\Delta t adjusted so α\alpha stays fixed as Δx\Delta x changes) measure the spreading rate ⟨x2⟩/t\langle x^2\rangle/t of a point release of heat: 4.0128×10−3 m2 s−14.0128\times10^{-3}\,\mathrm{m^2\,s^{-1}} at Δx=0.020 m\Delta x=0.020\,\mathrm{m} and 4.0032×10−3 m2 s−14.0032\times10^{-3}\,\mathrm{m^2\,s^{-1}} at Δx=0.010 m\Delta x=0.010\,\mathrm{m}. (a) The exact one-dimensional result is ⟨x2⟩/t=2D\langle x^2\rangle/t = 2D. Compute the absolute error of each run against this exact value. (b) Compute the ratio of the two errors (coarser over finer) and state what order of accuracy in space this confirms. (c) Someone argues: "matching ⟨x2⟩/t\langle x^2\rangle/t at one time, to this accuracy, verifies the scheme." Explain what stronger test the similarity collapse provides and why it would catch an error this single-time, single-moment check would miss.

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