106
5 2D Shallow-Water Modelling
• The upstream scheme follows from Ψ = 0. We have used this scheme in the
prediction of sea-level elevation in previous exercises.
• Ψ = 1 gives the Lax-Wendroff scheme.
• Ψ(r ) = max {0, min(2r, 1), min(r, 2)} define the so-called Superbee scheme.
In addition to this, we consider the Super-C scheme that is boundless by using
the Courant number in the denominator. This scheme is define by:
Ψ(r, |C|) =
⎧
⎨
⎩
min (2r/ |C| , 1)
: 0 ≤ r ≤ 1
min (r/2/(1 − |C|), r ) :
r > 1
0
: otherwise
where the Courant number C is calculated at the right-hand face of a control volume.
5.6.4 Stability Criterion for the Advection Equation
The stability criterion for the above explicit forms of the advection equation is:
C =
Δt
Δx
u ≤ 1
(5.27)
where C is the Courant number, and u is the f ow speed. Accordingly, time steps
have to satisfy the condition:
Δt ≤
Δx
u
(5.28)
Note that this is also a CFL condition, but this time based on fl w speed instead
of phase speed of waves. Conditions of C < 1 will always lead to a certain level of
numerical diffusion. This is because the displacement distance per time step is less
the grid spacing, so that some averaging will take place. Nevertheless, a particular
advection scheme might perform better than others, which will be investigated in
the following.
5.7 Exercise 11: Eulerian Advection
5.7.1 Aim
The aim of this exercise is to simulate the movement of non-buoyant Eulerian tracer
subject to the steady-state lake’s circulation predicted in Exercise 10. Different TVD
advection schemes will be tested.
5 2D Shallow-Water Modelling
• The upstream scheme follows from Ψ = 0. We have used this scheme in the
prediction of sea-level elevation in previous exercises.
• Ψ = 1 gives the Lax-Wendroff scheme.
• Ψ(r ) = max {0, min(2r, 1), min(r, 2)} define the so-called Superbee scheme.
In addition to this, we consider the Super-C scheme that is boundless by using
the Courant number in the denominator. This scheme is define by:
Ψ(r, |C|) =
⎧
⎨
⎩
min (2r/ |C| , 1)
: 0 ≤ r ≤ 1
min (r/2/(1 − |C|), r ) :
r > 1
0
: otherwise
where the Courant number C is calculated at the right-hand face of a control volume.
5.6.4 Stability Criterion for the Advection Equation
The stability criterion for the above explicit forms of the advection equation is:
C =
Δt
Δx
u ≤ 1
(5.27)
where C is the Courant number, and u is the f ow speed. Accordingly, time steps
have to satisfy the condition:
Δt ≤
Δx
u
(5.28)
Note that this is also a CFL condition, but this time based on fl w speed instead
of phase speed of waves. Conditions of C < 1 will always lead to a certain level of
numerical diffusion. This is because the displacement distance per time step is less
the grid spacing, so that some averaging will take place. Nevertheless, a particular
advection scheme might perform better than others, which will be investigated in
the following.
5.7 Exercise 11: Eulerian Advection
5.7.1 Aim
The aim of this exercise is to simulate the movement of non-buoyant Eulerian tracer
subject to the steady-state lake’s circulation predicted in Exercise 10. Different TVD
advection schemes will be tested.