The concentration of a species in an autocatalytic reaction is modelled as

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Computational Physics, Ch Numbers a machine can hold, and the two errors that follow

Problem

c(t) = c_0 e^{2t}

, with

c_0 = 1\,\mathrm{mol\,L^{-1}}

and

t

in seconds. The instantaneous rate

c'(t)

is to be estimated from sampled concentrations by the forward-difference formula

[c(t+h)-c(t)]/h

. (a) State the binary64 layout (the numbers of sign, exponent and stored significand bits) and use it to derive

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

, giving its decimal value to seven significant figures. (b) Expand

c(t+h)

as a Taylor series about

t

, subtract

c(t)

and divide by

h

to obtain the general form

[c(t+h)-c(t)]/h = c'(t) + \tfrac{1}{2}h c''(t) + \mathcal{O}(h^2)

, and hence write down the predicted truncation error of the forward difference at

t=0

as a function of

h

. (c) Evaluate that prediction at

h = 0.1\,\mathrm{s}

0.05\,\mathrm{s}

and

0.025\,\mathrm{s}

, then compute the forward-difference estimate itself at each of these three steps and its true error against

c'(0)

. Tabulate the ratio of successive true errors and state the order of the method that the ratios support.

Browse the problem library