42

EG2 for Compressible Flow

The 2nd Law cannot be derived from purely mechanical laws. It carries
the stamp of the essentially statistical nature of heat. (Bergman in Basic
Theories of Physics 1951)

The theory (quantum mechanics) yields a lot, but it hardly brings us closer
to the secret of the Old One. In any case I am convinced that He does not
throw dice. (Einstein to Born 1926)

Shut up and calculate. (Dirac on quantum mechanics)

42.1 G2 for the Compressible Euler Equations

We now present EG2 for the compressible Euler equations written in a system
as follows: Find ˆ

u(x, t) such that

R(ˆ

u)

≡

∂ ˆ

u

∂t

+

3

i=1

f

i

(ˆ

u)

,i

= 0,

in Q,

u(x, t)

· n(x, t) = 0,

in Γ

× I,

ˆ

u(

·, 0) = ˆu

0

,

in Ω,

(42.1)

where

ˆ

u = ρ

⎡
⎢

⎢

⎢

⎢

⎣

1

u

1

u

2

u

3

e/ρ

⎤
⎥

⎥

⎥

⎥

⎦

, f

i

(ˆ

u) = u

i

ˆ

u + p

⎡
⎢

⎢

⎢

⎢

⎣

0

δ

1i

δ

2i

δ

3i

u

i

⎤
⎥

⎥

⎥

⎥

⎦

,

with δ

ii

= 1, and δ

ij

= 0 if i

= j.

EG2 takes the general form: Find ˆ

u

∈ V

h

such that for all ˆ

v

∈ V

h

((R(ˆ

u), ˆ

v)) + ((hR(ˆ

u), R

u

(ˆ

v)))

+ ((ˆ

ν

∇u

i

,

∇v

i

)) + (u(

·, 0) − u

0

, v(

·, 0)) = 0,

(42.2)

376

42 EG2 for Compressible Flow

where V

h

is a finite element space of mesh size h in space-time of functions

ˆ

v with velocity components v satisfying the boundary condition v

· n = 0 on

Γ , ((

·, ·)) and (·, ·) represent L

2

(Q) and L

2

(Ω) scalar products, R

u

(ˆ

v) is the

linearization of R(ˆ

u) obtained by freezing the convective velocity at u noting

that R

u

(ˆ

u) = R(ˆ

u), and ˆ

ν = h

2

|R(ˆu)| is a shock-capturing artificial viscosity.

EG2 combines a weak satisfaction of the Euler equations with a weighted least
squares control of the residual R(ˆ

u) and thus represents a midway between

the Scylla of weak solution and Carybdis of least squares strong solution.

42.2 EG2 Satisfies the 2nd Law

We can show in the same way as we proved the 2nd Law for incompressible
flow in Chapter 18 that EG2 satisfies a weak form of the 2nd Law expressed in
any of the local forms (41.2)-(41.4) or global form (41.5). We give the detailed
proof in Body & Soul Vol 5 on Computational Thermodynamics, with the
prototype of the proof presented in [69].

We conclude that EG2 produces approximate solutions to the Euler equa-

tions which are irreversible in the general case of shocks and turbulence. Thus
EG2 produces approximate solutions which are irreversible, to be compared
with exact solutions, which would have been reversible had they existed, but
they don’t.

42.3 EG2 and the Classical Entropy

The classical entropy S can be obtained from the defining relation T dS =
dT + pdV to get S = log(T ρ

1

−γ

). The 2nd Law can be expressed in classical

form as ˙

S +

∇ · (wS) ≥ 0, and we can prove that EG2 weakly satisfies the

2nd Law in this form. As indicated, we prefer not to introduce S, since its
physical meaning is unclear. Nevertheless, we can as in [69] obtain choosing
as test function v in (42.2) an interpolant of the gradient

−S



(ˆ

u) of

−S(ˆu)

with respect to ˆ

u, the following global bound

−



Ω

S(ˆ

u(T ))dx

− ((hR(ˆu), R(ˆu)S



))

≤



Ω

−S(ˆu(0))dx,

where

−S



(ˆ

u) > 0 is the Hessian of

−S(ˆu), which gives a weighted bound

of the least squares term ((hR(ˆ

u), R(ˆ

u))) in terms of logarithms effectively

being bounded. The proof that EG2 weakly satisfies ˙

S +

∇(wS) ≥ 0 is similar

multiplying by

−S



(ˆ

u)φ with φ a non-negative test function. We can thus use

the classical entropy to obtain an a priori global bound for the stabilization
term, but that is the only useful role of the entropy we see. Of course a
posteriori we may check if the stabilization term is bounded, so in practice
the a priori bound may not be needed, and thus we may entirely forget about
S and expect that Nature does the same, since a sensor for S seems to be
missing.