Chapter 8 Configuring Fluidity - The Applied Modelling and ...

34 Numerical discretisation
where the hatted symbols indicate that the quantity in question is continuous between elements. In
the following sections, û will be used to indicate the flux velocity, which will have been calculated
with one of these methods. Note that using the averaging method, û = u on the interior of each
element with only the inter-element flux differing from u while for the projection method, û and u
may differ everywhere.
Upwind Flux In this case, the value of c at each quadrature point on the face is taken to be the
upwind value. For this purpose the upwind value is as follows: if the flow is out of the element then
it is the value on the interior side of the face while if the flow is into the element, it is the value on
the exterior side of the face. If we denote the value of c on the upwind side of the face by cupw then
equation (3.27) becomes:
�
ϕ
e
∂c
�
− (∇ · ϕ û) c + ϕn · û cupw = 0,
∂t ∂e
(3.29)
Summing over all elements, including boundary conditions and writing cint to indicate flux terms
which use the value of c from the current element only, we have:
�
��
ϕ
e e
∂c
�
− (∇ · ϕ û) c +
∂t ∂e∩∂Ω Dc
n · û gD
−
�
+
∂e∩∂ΩNc ∩∂Ω Dc
n · û cint
+
�
�
(3.30)
+
n · û cupw = 0.
∂e\(∂ΩD∩∂ΩN )
Second integration by parts In Fluidity the advection term with upwinded flux may be subsequently
integrated by parts again within each element. As this is just a local operation on the continuous
pieces of c within each element, the new boundary integrals take the value of c on the inside
of the element, cint. On the outflow boundary of each element this means the cint cancels against
cupw (when the summation over all elements occurs). Writing ∂e− for the inflow part of the element
boundary, we therefore obtain:
�
��
ϕ
e e
∂c
�
− ϕ û · ∇c +
∂t
�
+
∂e−∩∂Ω Dc
−
n · û (gD − cint)
n · û (cupw − cint)
∂e−\(∂ΩD∩∂ΩN )
�
= 0.
(3.31)
The difference cint − cupw on the inflow boundary remains, and is often referred to as a jump condition.
Note also that the boundary terms on the Neumann domain boundary and the outflow part of the
Dirichlet boundary (really also a Neumann boundary) have disappeared. Note that (3.30) and (3.31)
are completely equivalent. The advantage of the second form, referred to in Fluidity as “integratedby-parts-twice”,
is that the numerical evaluation of the integrals (quadrature), may be chosen not
to be exact (incomplete quadrature). For this reason the second form may be more accurate as the
internal outflow boundary integrals are cancelled exactly.
Local Lax-Friedrichs flux The local Lax-Friedrichs flux formulation is defined in Cockburn and Shu
[2001, p208]. For the particular case of tracer advection, this is given by:
�n · u c = 1
2 n · û (cint + cext) − C
2 cint − cext
(3.32)
where cext is the value of c on the exterior side of the element boundary and in which for each facet
s ⊂ ∂e:
C = sup |û · n| (3.33)
x∈s