Modeling Concepts - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 233

6.2.3 N C A

UMERICAL ALCULATION OF DVECTION

6.2.3.1 Spatial Approach

Three common approaches are used to estimate concentration values at control
volume faces:

• Linear approach
• Upstream stepwise approach
• Quadratic upwind approach (QUICK)

6.2.3.1.1 Linear Approach

In the linear approach it is assumed that:

Cx ∆ +

+

C ∆ x

C =

i i − 1 i − 1 i

i

−

1 2

∆ x ∆ x

i − 1 i

Assuming a discretization where the grid size is uniform, it is easily seen that this
approach generates central differences as obtained using the Taylor series (see
Section 6.2.4).

6.2.3.1.2 Upstream Stepwise Approach

In this case, it is assumed that the concentration at the left face is

( )

Q >⇒ 0 C = C

i i i − 1

−

1 2

( )

Q <⇒ 0 C = C

i i i

−

1 2

This discretization respects the transportivity property of advection. This property
states that advection can transport properties only downstream or that information
comes only from upstream. The linear approach does not respect this property
because volume

will get information of downstream concentration through the
average process. The violation of this property can generate instabilities and will
create conditions to obtain negative values of the concentration. The upstream
discretization avoids this limitation but, as shown in the following paragraphs, it can
introduce unrealistic numerical diffusion.

i

6.2.3.1.3 Quadratic Upwind Approach (QUICK)

The quadratic upwind approach, or QUICK scheme, is an attempt at a compromise
between respecting the transportivity property and keeping numerical diffusion at
low values. In this case, it is assumed that the concentration distribution around a
point follows a quadratic distribution centered on the upstream side of the face