The theta method for is y_ n+1 = y_n + h [(1 - ) ,f(t_n

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Ordinary differential equations: order, stability and stiffness

Problem

The theta method for

y' = f(t, y)

y_{n+1} = y_n + h\bigl[(1 - \theta)\,f(t_n, y_n) + \theta\,f(t_{n+1}, y_{n+1})\bigr],

where

\theta

is a constant with

0 \le \theta \le 1

. It is Euler's method for

\theta = 0

, backward Euler for

\theta = 1

and the trapezoidal rule for

\theta = \tfrac12

. For an implicit method the local truncation error is defined as in the chapter, with the exact

y(t_{n+1})

in place of

y_{n+1}

on the right-hand side. ((a)) Show that

\tau = (\tfrac12 - \theta)\,h\,y'' + O(h^2)

. For which

\theta

is the method of order 2, and what is the leading term of

\tau

then? ((b)) Apply the method to

y' = -y

with

y(0) = 1

, all dimensionless. Find the global error (computed minus exact) at

t = 1

for

\theta = 0

\tfrac12

and

1

, each with

h = 0.1

and with

h = 0.05

. ((c)) Estimate the same errors from

\tau

by the accumulation argument of the chapter. Each step commits an error

h\tau

, which lowers the computed value below the exact one by that amount, and an error made at time

s

is multiplied by

e^{-(t - s)}

by time

t

. Compare with (b), and say by what factor each error falls when

h

is halved.

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