Chapter 1's drag-free explicit Euler integrator for a body falling under gravity…

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 for a body falling under gravity updates the height

y

(above the ground) and downward speed

v

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

starting from

v_0=0

and a stated

y_0

, with

g=9.8\,\mathrm{m\,s^{-2}}

. The quantity conserved by the true physics, per unit mass, is

E_n=\tfrac12v_n^2+gy_n

. A stone is dropped from

y_0=800\,\mathrm{m}

and its fall is followed to

t=10\,\mathrm{s}

. (a) Show that a single Euler step always changes

E

by exactly

\Delta E_{\rm step}=\tfrac12 g^2(\Delta t)^2

, whatever the speed

v_n

at that step. (b) Hence show that after

N=t/\Delta t

steps the fractional energy drift is

\dfrac{\Delta E}{E_0}=\dfrac{g\,\Delta t\,t}{2y_0}

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

\Delta t=0.4\,\mathrm{s}

and at

\Delta t=0.05\,\mathrm{s}

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

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