A metal rod is discretised with grid spacing and diffusivity

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Numerical and computational mathematics

Error, Chance and the Machine, Ch Partial differential equations: stability, convergence and solvers

Problem

A metal rod is discretised with grid spacing Δx=0.01 m\Delta x = 0.01\,\mathrm{m} and diffusivity D=1.0×10−4 m2 s−1D = 1.0\times10^{-4}\,\mathrm{m^2\,s^{-1}}. The initial temperature (above ambient) alternates +1∘C,−1∘C,+1∘C,…+1^\circ\mathrm{C}, -1^\circ\mathrm{C}, +1^\circ\mathrm{C}, \ldots between neighbouring grid points, a sharp sawtooth. One step of Crank-Nicolson is taken with Δt=50 s\Delta t = 50\,\mathrm{s}, giving α=DΔt/Δx2=50\alpha = D\Delta t/\Delta x^2 = 50. After the step, the amplitude of the sawtooth has fallen only slightly, by a factor of about 0.980.98. Someone argues: "Crank-Nicolson is unconditionally stable for any α\alpha, so this step size is safe, and since the amplitude stayed bounded and only shrank, this must be an accurate solution of the diffusion equation after 50 seconds." Find the error in this argument. ((a)) Compute the amplification factor G(π)G(\pi) for the grid-scale mode at α=50\alpha = 50. ((b)) Compute the true physical decay factor over the same 50 s50\,\mathrm{s} for a Fourier mode of wavelength 2Δx2\Delta x, using exp⁡(−Dk2t)\exp(-Dk^2t) with k=π/Δxk = \pi/\Delta x. ((c)) State the actual error in the reasoning.

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