A rod has both ends held at ambient temperature and is discretised with four interior…

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 rod has both ends held at ambient temperature and is discretised with four interior grid points, numbered 1,…,41,\ldots,4. Let ujnu_j^n be the temperature above ambient at point jj after nn steps, with u0n=u5n=0u_0^n = u_5^n = 0. Take the diffusion number α=DΔt/Δx2=1\alpha = D\Delta t/\Delta x^2 = 1 and the initial temperatures u0=(0,1,1,0) K\mathbf{u}^0 = (0, 1, 1, 0)\,\mathrm{K}. ((a)) Write out the Crank-Nicolson step (1−α2T)u1=(1+α2T)u0\bigl(\mathsf{1} - \tfrac{\alpha}{2}\mathsf{T}\bigr)\mathbf{u}^1 = \bigl(\mathsf{1} + \tfrac{\alpha}{2}\mathsf{T}\bigr)\mathbf{u}^0 in full for these data. Solve it by the tridiagonal elimination of Chapter 2, and give the four pivots and u1\mathbf{u}^1. ((b)) Take one explicit step, u1=(1+αT)u0\mathbf{u}^1 = (\mathsf{1} + \alpha\mathsf{T})\mathbf{u}^0, from the same data at the same α\alpha. ((c)) The grid-scale mode of this rod is sin⁡(jθ4)\sin(j\theta_4) with θ4=4π/5\theta_4 = 4\pi/5. Find the factor by which each scheme multiplies it in one step, and say which of your two results for u1\mathbf{u}^1 can be believed.

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