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Euler Equations (FV)
• They are written in the form:
∂U
+∇ iFU
( ) = 0
∂t
U { ρρ , u
, ρE} (conserved variables)
FU ( ) { ρu, ρuu + pI,( ρE+
pu ) } (fluxes)
• The discrete conserved variables are integral
means over the generic control volume Ω
n
U
1
= U dV Ω
∫ Ω
∂
• The weak formis UdV=−
[ nFU ( )] dS
• The solution is advanced in time by
the first-order explicit scheme
(approximate Riemann solver)
n+ 1 n+
1
∂t ∫ ∫ i
Ω
∂Ω
n
n
Ω U − Ω U =∆∑ t [ niF( U)]
∂Ω
• MUSCL-like technique yields 2 nd -order accurate numerical fluxes
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Finite Volume Peculiarities
• All quantities (velocities and pressures) are discretized at
nodes
• Physical status of FV depends only upon its volume, not
on its shape (OK for fluid)
• However, geometry is still used to compute volume and
fluxes
• Transport (among FVs, i.e. nodes) is
computed internally to each FE : no
need to know neighbor FEs
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