Chapter 1's drag-free explicit Euler integrator updates height and downward speed by…

PHYSICS · Subjective · Fermi University Physics · Computational Physics · Research software practice, verification and validation · Simulating classical dynamical systems

Computational Physics, Ch A workspace that can be rerun next year

Problem

Chapter 1's drag-free explicit Euler integrator updates height

y

and downward speed

v

v_{n+1}=v_n+g\Delta t,\qquad y_{n+1}=y_n-v_n\Delta t,

from

v_0=0

and a stated

y_0

, with conserved quantity per unit mass

E_n=\tfrac12v_n^2+gy_n

. A lunar lander's uncontrolled fall is simulated using the Moon's surface gravity

g=1.62\,\mathrm{m\,s^{-2}}

, dropped from

y_0=300\,\mathrm{m}

and followed to

t=15\,\mathrm{s}

, while it is still well above the surface. (a) Using the fact that a single Euler step changes

E

by exactly

\tfrac12g^2(\Delta t)^2

regardless of the speed at that step, show that the fractional energy drift after time

t

\Delta E/E_0=g\Delta t\,t/(2y_0)

. (b) Evaluate this fractional drift, as a percentage, at

\Delta t=0.2\,\mathrm{s}

and at

\Delta t=0.025\,\mathrm{s}

. (c) State the ratio of the two drifts and compare it with the ratio of the two step sizes.

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