A thin steel rod obeys the one-dimensional diffusion equation for temperature,

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 thin steel rod obeys the one-dimensional diffusion equation for temperature, ∂c/∂t=D ∂2c/∂x2\partial c/\partial t = D\,\partial^2 c/\partial x^2, with thermal diffusivity D=4.0×10−5 m2 s−1D = 4.0\times10^{-5}\,\mathrm{m^2\,s^{-1}}. It is discretised on a grid with spacing Δx=2.0 mm\Delta x = 2.0\,\mathrm{mm} and stepped forward with the explicit scheme cjn+1=αcj−1n+(1−2α)cjn+αcj+1nc_j^{n+1} = \alpha c_{j-1}^n + (1-2\alpha)c_j^n + \alpha c_{j+1}^n, α=DΔt/Δx2\alpha = D\Delta t/\Delta x^2, using a time step Δt=0.050 s\Delta t = 0.050\,\mathrm{s}. (a) Compute α\alpha and say whether the stability condition is satisfied. (b) At one instant three neighbouring nodes read cj−1n=80.0∘Cc_{j-1}^n = 80.0^\circ\mathrm{C}, cjn=60.0∘Cc_j^n = 60.0^\circ\mathrm{C}, cj+1n=50.0∘Cc_{j+1}^n = 50.0^\circ\mathrm{C}. Find cjn+1c_j^{n+1}.

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