An aluminium rod of length has thermal diffusivity

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

An aluminium rod of length

L = 0.02\,\mathrm{m}

has thermal diffusivity

D = 9.7\times10^{-5}\,\mathrm{m^2\,s^{-1}}

. Its ends are held at fixed temperatures and it is discretised into

N = 40

interior grid points, uniformly spaced. The semi-discrete system

\dot{\mathbf{u}} = (D/\Delta x^2)\mathsf{T}\mathbf{u}

for this grid has eigenvalues

\lambda_k = -\dfrac{4D}{\Delta x^2}\sin^2\!\left(\dfrac{k\pi}{2(N+1)}\right)

k = 1, \ldots, N

, where

\Delta x = L/(N+1)

. ((a)) Using the ratio

|\lambda_N|/|\lambda_1|

of the largest to the smallest eigenvalue magnitude (Chapter 12's measure of stiffness for a system of ordinary differential equations), estimate the stiffness ratio of this grid. ((b)) Find the explicit stability limit

\Delta t_{\max}

and the number of explicit steps needed to simulate

T = 5\,\mathrm{s}

of cooling. ((c)) Instead, Crank-Nicolson is run with

\Delta t = 0.01\,\mathrm{s}

. Using the fact that the tridiagonal (Thomas) solver of Chapter 2 needs about

8

arithmetic operations per grid point per step, find the number of Crank-Nicolson steps and the total number of arithmetic operations spent in the tridiagonal solves over the same

5\,\mathrm{s}

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