A dye diffuses in still water with diffusion coefficient

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 dye diffuses in still water with diffusion coefficient D=1.0×10−9 m2 s−1D = 1.0\times10^{-9}\,\mathrm{m^2\,s^{-1}}. Its concentration c(x,t)c(x,t) is discretised on a grid with spacing Δx=0.50 mm\Delta x = 0.50\,\mathrm{mm} and advanced 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 using a time step Δt=100 s\Delta t = 100\,\mathrm{s}. (a) Compute α\alpha and confirm the run is stable. (b) At some instant three neighbouring nodes read cj−1n=0.20 mol m−3c_{j-1}^n = 0.20\,\mathrm{mol\,m^{-3}}, cjn=0.50 mol m−3c_j^n = 0.50\,\mathrm{mol\,m^{-3}}, cj+1n=0.10 mol m−3c_{j+1}^n = 0.10\,\mathrm{mol\,m^{-3}}. Find cjn+1c_j^{n+1}.

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