An Euler-Cromer routine integrates a simple harmonic oscillator with , ,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Research software practice, verification and validation

Scientific Computing for Physicists, Ch Is the equation solved right? Testing, debugging and code verification

Problem

An Euler-Cromer routine integrates a simple harmonic oscillator with

\omega = 1\,\mathrm{rad\,s^{-1}}

x(0)=1\,\mathrm{m}

v(0)=0\,\mathrm{m\,s^{-1}}

, exact solution

x(t)=\cos(\omega t)

. ((a)) A unit test runs the routine to

t=1\,\mathrm{s}

with

h=0.001\,\mathrm{s}

. State the exact value

x(1)=\cos(1)

to six significant figures, and say whether the test passes a tolerance of

2\times10^{-3}\,\mathrm{m}

, given a measured error of

4.2\times10^{-4}\,\mathrm{m}

at this step. ((b)) The Euler-Cromer map is area-preserving: the Jacobian determinant of one step is exactly

1

for any

h

. A property test computes this determinant numerically, by LU factorisation at

h=0.01\,\mathrm{s}

, and checks

|\det - 1|

against tolerance

5\epsilon_{\rm mach}

, with

\epsilon_{\rm mach}=2^{-52}

. Compute the tolerance, and say whether a measured value of

1.1\times10^{-16}

passes it. ((c)) An order-of-accuracy test gives maximum errors

e(0.02)=8.43\times10^{-3}\,\mathrm{m}

e(0.01)=4.21\times10^{-3}\,\mathrm{m}

and

e(0.005)=2.10\times10^{-3}\,\mathrm{m}

. Compute

\hat{p}

from each consecutive pair using

\hat{p}=\ln[e(h)/e(h/2)]/\ln2

, and say whether the code is verified against Euler-Cromer's formal order

p_{\rm formal}=1

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