The forward-time centred-space (FTCS) diffusion scheme is run with

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

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

Problem

The forward-time centred-space (FTCS) diffusion scheme is run with D=2.0×10−5 m2 s−1D = 2.0\times10^{-5}\,\mathrm{m^2\,s^{-1}}, Δx=5.0×10−3 m\Delta x = 5.0\times10^{-3}\,\mathrm{m} and Δt=0.80 s\Delta t = 0.80\,\mathrm{s}. Its von Neumann amplification factor is G(θ)=1−2α(1−cos⁡θ),α=DΔtΔx2.G(\theta) = 1 - 2\alpha(1-\cos\theta), \qquad \alpha = \frac{D\Delta t}{\Delta x^2}. ((a)) Compute α\alpha for the stated grid and step. ((b)) Find G(θ)G(\theta) at θ=π\theta = \pi (the grid-scale mode) and state whether the scheme is stable at this step.

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