A steel bar of length is divided into cells of width

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

A steel bar of length 0.12 m0.12\,\mathrm{m} is divided into N=6N = 6 cells of width Δx=0.02 m\Delta x = 0.02\,\mathrm{m}, with a node at the centre of each cell, and both ends are insulated (zero heat flux). The thermal diffusivity is D=1.0×10−5 m2 s−1D = 1.0\times10^{-5}\,\mathrm{m^2\,s^{-1}}. A zero-flux end is implemented by giving the missing neighbour outside each end the end node's own value (a ghost point mirrored across the end face), which changes the diagonal entry of the second-difference matrix T\mathsf{T} at each boundary row from −2-2 to −1-1; all other rows keep the usual [1,−2,1][1,-2,1] pattern. For this boundary condition the eigenvalues of T\mathsf{T} are μj=−4sin⁡2 ⁣(jπ2N),j=0,1,…,N−1.\mu_j = -4\sin^2\!\left(\frac{j\pi}{2N}\right), \qquad j = 0, 1, \dots, N-1. ((a)) Evaluate μj\mu_j for j=0j = 0 and for j=N−1=5j = N-1 = 5, and state which of these is the largest in magnitude. ((b)) The semi-discretised system is u˙=(D/Δx2)Tu\dot{\mathbf{u}} = (D/\Delta x^2)\mathsf{T}\mathbf{u}. Using the explicit Euler stability region from Chapter 12, find the largest time step Δt\Delta t for which the explicit scheme is stable.

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