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Time integration (3)
Some pseudo-viscosity is needed to stabilize solution at shock fronts:
⎧⎪
ρ ∇⋅ − ∇⋅ ∇⋅ <
q′ = ⎨
⎪⎩ 0 for ( ∇⋅v) ≥0
2 2
[ Cl
Q
( v) Cla
L
( v)] for ( v) 0
C , C quadratic and linear coefficient
Q
L
l characteristic length of the element
p
q=
min ( q′ , )
2
a dilatational wave speed
4. Former expression may be approximated to the first order by:
( ρiV ) −( ρiV )
n V −V
=− ( p+
q)
∆t
∆t
L n L n
L n n
5. By noting that ( ρV) = ( ρV)
= M we obtain for the internal energy:
i = i − ( p+
q)
L n n
L
V −V
n
M
n
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Time integration (4)
L
6. The obtained i is just a first guess since the pressure changes over
the step and must satisfy the state equation!
7. Obtain L-value implicitly by iterating the following expressions
(just one or two iterations are usually sufficient):
p L = f( ρ
L , i
L ) f suitable state equation
n L L n
L n p + p n V −V
i = i − ( + q )
n
2 M
8. Compute true end-of-step configuration:
x = x +∆t⋅w
n+ 1 n n+
1/2
9. New volume:
n 1
V +
18
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