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ARTIFICIAL VISCOSITY MODELS
– 4 th order operator : it is a less common operator. It acts as a bi-Laplacian and is mainly used to
control spurious high-frequency wiggles.
The way they are combined is determined both by the sensor and by user-defined parameters (smu2 and
smu4 in the run.dat file).
Both operator contributions are first computed on each cell vertex, and are then scattered back to nodes
(there is no divergence here, as it is done directly during the scattering operation).
The 2 nd order operator
A cell contribution of the 2 nd order AV is first computed on each vertex of the cell Ω j :
R k∈Ωj = − 1 N v
V Ωj
∆t Ωj
smu2 ζ Ωj (w Ωj − w k )
The nodal residual is then found by adding the surrounding cells contributions :
(A.12)
dw k = ∑ j
R k∈Ωj
(A.13)
For example, on a 1D uniform mesh, of mesh size ∆x, and for ζ Ωj = ζ = cste :
which can be interpreted as :
with :
ν AV =
smu2 ζ ∆x2
2∆t
=
dw k = − smu2 ζ
2
∆x
∆t (w k−1 − 2w k + w k+1 )
(A.14)
∫
dw k = −ν AV (∆ k,∆x w) dx (A.15)
smu2 ζ ∆x|u + c|
2 CFL
and
∆ F D
k,∆x w = w k−1 − 2w k + w k+1
∆x 2
(A.16)
where ∆ F k,∆x D is exactly the classical FD Laplacian operator evaluated at k and of size ∆x.
This shows that ν AV can be seen as an “artificial” viscosity (it has the same units as a physical viscosity),
which is controlled by the user-defined parameter smu2. The smu2 parameter is therefore dimensionless.
A.3.1
The 4 th order operator
The technique used for the 4 th order operator is identical to the technique of the 2 nd order operator.
A cell contribution is first computed on each vertex :
R k∈Ωj = 1 V Ωj
[
smu4 (
N v ∆t ⃗ ]
∇w) Ωj · (⃗x Ωj − ⃗x k ) − (w Ωj − w k )
Ωj
The nodal value is then found by adding every surrounding cells contributions :
(A.17)
dw k = ∑ j
R k∈Ωj
(A.18)
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