246
5. Finite-Volume Methods
5.1.2 Entropy-Consistent Solutions
It is therefore preferable to obtain alternative criteria for selecting the physically
relevant weak solution. These criteria may be derived directly from physical principles. Stoker (1957) eliminated nonphysical shocks in shallow-water flow by requiring that "the water particles do not gain energy upon crossing a shock front."
In gas dynamics, thennodynamic principles require that entropy be nondecreasing
at the shock. Generalized entropy conditions can also be derived for any system of
one or two scalar conservation laws of the form (5.6) by considering the limiting
behavior of the corresponding viscous system as the viscosity approaches zero
(Lax 1971).
For example, a generalized entropy function for the inviscid Burgers's equation
01/1
ot
+
ox
(1/12) =0
2
(5.11)
is 1/12. When 1/1 is a weak solution to Burgers 's equation, 1/12 is a weak solution to
the inequality
01/12 +
ot ox
(21/13) < O.
3
-
(5.12)
If the solutions to Burgers's equation are differentiable, the left side of (5.12) is
identically zero and the time rate of change of the integral of 1/12 over any spatial
domain is equal to the divergence of the entropy flux, 21/13/3, through the edges
of the domain. But if the solution of Burgers 's equation is discontinuous, (5.12)
can no longer be satisfied by an equality. The sense of the inequality demanded
by (5.12) is that which matches the limiting behavior of 1/12 for solutions to the
viscous Burgers's equation
if x:::: 0,
otherwise.
01/1 +
ot
as E -)- 0 (LeVeque 1992, p. 37).
ox
(1/1
2)
21/1
= E 0
ox 2
2
Consider two possible weak solutions to the inviscid Burgers's equation (5.11) ,
both of which are consistent with the initial condition
1/1 (x, 0) =
which is a speed of i. The second solution, shown in Fig . 5.3b, is the rarefaction
The first solution, shown in Fig. 5.3a, consists of a unit-amplitude downward!
jump moving to the right at the speed given by the Rankine-Hugoniot condition,
1The jump is "downward" in the sense that the fluid level drops during the passage of the discon -
tinu ity.
5. Finite-Volume Methods
5.1.2 Entropy-Consistent Solutions
It is therefore preferable to obtain alternative criteria for selecting the physically
relevant weak solution. These criteria may be derived directly from physical principles. Stoker (1957) eliminated nonphysical shocks in shallow-water flow by requiring that "the water particles do not gain energy upon crossing a shock front."
In gas dynamics, thennodynamic principles require that entropy be nondecreasing
at the shock. Generalized entropy conditions can also be derived for any system of
one or two scalar conservation laws of the form (5.6) by considering the limiting
behavior of the corresponding viscous system as the viscosity approaches zero
(Lax 1971).
For example, a generalized entropy function for the inviscid Burgers's equation
01/1
ot
+
ox
(1/12) =0
2
(5.11)
is 1/12. When 1/1 is a weak solution to Burgers 's equation, 1/12 is a weak solution to
the inequality
01/12 +
ot ox
(21/13) < O.
3
-
(5.12)
If the solutions to Burgers's equation are differentiable, the left side of (5.12) is
identically zero and the time rate of change of the integral of 1/12 over any spatial
domain is equal to the divergence of the entropy flux, 21/13/3, through the edges
of the domain. But if the solution of Burgers 's equation is discontinuous, (5.12)
can no longer be satisfied by an equality. The sense of the inequality demanded
by (5.12) is that which matches the limiting behavior of 1/12 for solutions to the
viscous Burgers's equation
if x:::: 0,
otherwise.
01/1 +
ot
as E -)- 0 (LeVeque 1992, p. 37).
ox
(1/1
2)
21/1
= E 0
ox 2
2
Consider two possible weak solutions to the inviscid Burgers's equation (5.11) ,
both of which are consistent with the initial condition
1/1 (x, 0) =
which is a speed of i. The second solution, shown in Fig . 5.3b, is the rarefaction
The first solution, shown in Fig. 5.3a, consists of a unit-amplitude downward!
jump moving to the right at the speed given by the Rankine-Hugoniot condition,
1The jump is "downward" in the sense that the fluid level drops during the passage of the discon -
tinu ity.