Numerical Methods and Its Solution - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 212

unstable solution using the Euler method. Typically for explicit methods, numerical
stability can be improved by decreasing the step size. An alternative is to use an
implicit method which allows more flexibility in the step size selection. The implicit
method makes use of information at a future step, u t + 1 , as well as at u t. To illustrate
the structure of an implicit approach we present the one-step backward Euler method
which is given as

u t + 1 = u t + h × f ( t t + 1 , u t + 1 ) (7.73)

We notice that the implicit backward Euler method has the u t + 1 term on both the
left and right hand side of Eq. (7.73) and we don't have an explicit formula for u t + 1 .
Instead, we have to solve an equation to find the roots for u t + 1 . This step makes the
implicit scheme much more time consuming than the explicit Euler method and if
the differential equation is complicated, an implicit method can be very difficult to
use. Taking the same example we have been using we rewrite Eq. (7.73) as

1
τ ( u f

u t + 1 = u t + h × − u t + 1 ) (7.74)

=

and after rearranging and substituting in the values with h 0 . 00006, the first

increment is

h
τ u f

0.00001
0.000035

u t + 0 +

u t + 1 = = = 0.63158

h
τ

0.00001
0.000035

1 + 1 +

If we repeating for successive time steps and plot it against the explicit forward Euler
method, we can see that the implicit method is stable (Fig. 7.28 ).

Fig. 7.28 A comparison of
the solution based on the
implicit backward Euler
method with the explicit
forward Euler and the exact
analytical method. Time step
h = 0 . 00006 s

1.8

1.6

1.4

1.2

1.0

0.8

0.6

Forward Euler
Analytical
Backward Euler

0.4

0.2

0.0

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005

t

We have introduced some numerical methods for solving ODEs and it is hoped
that this will allow the reader to tackle some commonly used Lagrangian particle
tracking models used in many CFD packages. For the interested reader, any academic
textbook on numerical analysis will provide deeper insight into this field.