Numerical Methods and Its Solution - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 191

Fig. 7.22 Solutions for the
convection transport of φ

and rearranging

d 2 φ
dx 2

φ P − φ W dφ

dx −

x
2

O( x 2 )

= + (7.47)

x

If we substitute Eq. (7.47) into Eq. (7.46) we get
u dφ

dφ
dx

ux
2

d
dx

O( x 2 )

dx −

+ = 0 (7.48)

Equation (7.48) represents the continuous equation of the UD scheme that we have
applied using finite volume approach. The first term, u d dx is the convective transport
of φ and the remaining terms are in the form are the additional terms form the Taylor
series expansion which consists of the order error term. It can be seen that this term
resembles the diffusion term in the convection-diffusion Eq. (7.38). Therefore we
see that although we apply a discretisation on pure convection, the UD results in a
solution that has some diffusion occurring. This false diffusion occurs through the
numerical scheme as shown and is not a physical mechanism such as the diffusion
that occurs through viscosity. The false diffusion coefficient is proportional to the
control volume size and therefore refining the mesh will reduce the influence of the
false diffusion as was seen earlier (Fig. 7.22 ). While the UD scheme can provide
converging results under situations where Pe > 2, it is not entirely accurate given that
it is first order accurate and it is prone to false diffusion.

7.3 Numerical Solution

A system of linear or non-linear algebraic equations is produced from the discreti-
sation step, and this can be solved by some numerical methods. The complexity and
size of the set of equations depends on the dimensionality and geometry of the phys-
ical problem. In this section we present essentially two types of numerical methods:
direct methods and iterative methods .