A species diffuses and also decays by a first-order reaction

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 species diffuses and also decays by a first-order reaction, so its concentration obeys ∂c∂t=D∂2c∂x2−kc,\dfrac{\partial c}{\partial t} = D\dfrac{\partial^2 c}{\partial x^2} - kc, with DD in m2 s−1\mathrm{m^2\,s^{-1}} and decay rate kk in s−1\mathrm{s^{-1}}. The explicit scheme steps this forward as cjn+1=cjn+Δt[Dcj+1n−2cjn+cj−1nΔx2−kcjn].c_j^{n+1} = c_j^n + \Delta t\left[D\dfrac{c_{j+1}^n - 2c_j^n + c_{j-1}^n}{\Delta x^2} - kc_j^n\right]. (a) Write this update in the weighted-average form cjn+1=αcj−1n+w0cjn+αcj+1nc_j^{n+1} = \alpha c_{j-1}^n + w_0 c_j^n + \alpha c_{j+1}^n with α=DΔt/Δx2\alpha = D\Delta t/\Delta x^2, identify w0w_0, and use the argument that no weight in such an update may be negative to derive the resulting condition on Δt\Delta t. (b) A run uses D=5.0×10−6 m2 s−1D = 5.0\times10^{-6}\,\mathrm{m^2\,s^{-1}}, Δx=1.0 mm\Delta x = 1.0\,\mathrm{mm} and k=0.020 s−1k = 0.020\,\mathrm{s^{-1}}. Find the largest Δt\Delta t that condition allows, and compare it with the largest Δt\Delta t allowed on the same grid with the reaction term switched off (k=0k=0).

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