A pollutant diffuses vertically through soil 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 pollutant diffuses vertically through soil with D=2.0×10−10 m2 s−1D = 2.0\times10^{-10}\,\mathrm{m^2\,s^{-1}}. The concentration is discretised with grid spacing Δx=1.0 cm\Delta x = 1.0\,\mathrm{cm} and time step Δt=1.0 day=86400 s\Delta t = 1.0\,\mathrm{day} = 86400\,\mathrm{s}, using 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. (a) Compute α\alpha and confirm the stability condition holds. (b) At some instant three neighbouring nodes read cj−1n=5.0 mg L−1c_{j-1}^n = 5.0\,\mathrm{mg\,L^{-1}}, cjn=3.0 mg L−1c_j^n = 3.0\,\mathrm{mg\,L^{-1}}, cj+1n=1.0 mg L−1c_{j+1}^n = 1.0\,\mathrm{mg\,L^{-1}}. Find cjn+1c_j^{n+1}.

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