Quasilinear equationsdiffer from semilinear equations only in having the speedc depend additionally on the solution,
∂p
∂t +c(x,t,p)∂p
∂x =r(x,t,p). (2.37)
This makes the characteristic ODEs fully coupled, dP
dt =r(X,t,P), dX
dt =c(X,t,P). (2.38)
For semilinear equations, the equation (2.34a) for the characteristic curves x = X(t;A)decouples fromP, and standard existence and uniqueness results guarantee that two curves starting from different initial positions (and having different values ofP(t)) will never intersect. This is not the case for the coupled equations in (2.38), where characteristics can cross and hence predict several differentP(t;Aj)values occurring simultaneously at the samexposition.
As an example, consider theinviscid Burgers equation,
∂p
∂t +p∂p
∂x =0, (2.39)
with initial conditions
p(x,0)=
1− |x| |x| ≤1,
0 otherwise. (2.40)
Equation (2.39) is an important mathematical model that will arise again later in other contexts; one aspect of its importance can be observed by noting that if the propertypis the flow velocity, then (2.39) is the convective derivative of the velocity (2.5) (the Lagrangian form of the acceleration), DvDt =vt+vvx=0.
The characteristic equations for this example are dX
dt =P, X(0;A)=A, (2.41a)
dP
dt =0, P(0;A)=
1− |A| |A| ≤1,
0 else. (2.41b)
We note that for this problemPremains constant along each characteristic, so that X =Pt+Aand the solution can be expressed as
2.6 Shocks in Quasilinear Equations 35
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
X =A, P=0 A<−1, X =(1+A)t+A, P=1+A −1≤A≤0, X =(1−A)t+A, P=1−A 0≤A≤1,
X =A, P=0 A>1
(2.42)
and restated in an explicit form fort≥0
p(x,t)=
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
0 x<−1,
(1+x)/(1+t) −1≤x≤t, (1−x)/(1−t) t≤x≤1,
0 x>1.
(2.43)
There are several points to note about this representation of the solution—first, the piecewise-defined solutions are not mutually exclusive for t > 1. Consequently multiple values are being predicted forpat some locations whent >1. In relation to this, the third subcase becomes undefined at timet =1, changing from negative slopes fort<1 to positive slopes fort>1. Figure2.3shows the characteristic curves andp(x,t)profiles given by (2.43). We observe that the portion of the solution starting fromx∈ [−1,0]spreads out over an increasingly large region as its characteristics separate from each other (this is sometimes called anexpansion fanorrarefaction wave).
In contrast, the portion of the solution starting fromx∈ [0,1]is being compressed into a smaller region (up untilt =1) and is referred to as acompressive wave. Its slope steepens and overturns fort>1 to yield what could be described as a multi-valued
“breaking wave”.
Mathematically this part of the solution is predicting three values for a physical quantity (maybe a density, concentration or temperature, for example) that should have a unique value at any pointxat a given timet.
0 1 2
-2 -1 0 1 2
t
x -2 -1 0 x 1 2 3
1
0
Fig. 2.3 (Left) characteristic curvesX(t;A)given by (2.42) in thextplane and a 3D view of the evolving multi-valued solutionp(x,t)(2.43) (Right)
xs(t)
x 1 2
0 -1
1
0 p+(x, t)
p−(x, t)
x 1 2
0 -1
1
0
Fig. 2.4 (Left) The multivalued solution (2.43) and insertion of a shock atx = xs(t), (Right) reduction to a single-valued solution given by (2.46)
There is a systematic and rigorous approach for correcting this unphysical behav- iour and modifying the solution to make it single-valued through the use ofshocks.
Shocks are moving jump discontinuities in the solution that separate one piecewise- defined portion of the solution ahead of the shock,p+(x,t), from another part behind the shock, p−(x,t), eliminating the multi-valued behaviour (see Fig.2.4). Shock waves are common in many physical systems, including acoustics (sonic booms), fluid flow (hydraulic jumps), and traffic flow (moving traffic jams).
The construction of shock-corrected solutions builds on the idea that if a shock is inserted at one position,x=xs(t), it can appropriately separate the overlapping sets of characteristic curves to produce a well-defined single-valued solution everywhere away from the shock. The constructed solution should satisfy all of the properties expected for the conservation law and we use this to derive the equation for the motion of the shock position.
Consider a general quasilinear equation describing the transport of a property p(x,t)according to a flux functionq=q(p),
∂p
∂t +∂q(p)
∂x =0, (2.44)
which has been derived for smooth solutions via the Reynolds transport theorem from a conservation law forp. The integrated form of (2.44) on a fixed domaina≤x≤b includes contributions from fluxes at the ends of the domain,
d dt
b a
p dx
+q(p) x=b
x=a
=0. (2.45)
If a shock were inserted at some positionx=xs(t), the piecewise-defined form of the solution becomes
2.6 Shocks in Quasilinear Equations 37
p(x,t)=
p−(x,t) a≤x<xs(t),
p+(x,t) xs(t) <x≤b. (2.46) Separating (2.45) with respect to dependence on the solution to the left or right of the shock then yields
d dt
xs
a
p−dx
−q(p−(a,t))
+
d dt
b xs
p+dx
+q(p+(b,t))
=0.
(2.47) Applying Leibniz’s rule (2.8) then gives
xs a
∂p−
∂t dx+p−(xs,t)dxs
dt
+ b xs
∂p+
∂t dx−p+(xs,t)dxs
dt
+ [q(p+(b,t))−q(p−(a,t))] =0.
By adding and subtractingq(p±)(xs)and re-grouping terms, we obtain xs
a
∂tp−dx+q(p−(xs,t))−q(p−(a,t))
+ b
xs ∂tp+dx+q(p+(b,t))−q(p+(xs,t))
−q(p−(xs,t))+q(p+(xs,t))+ [p−(xs,t)−p+(xs,t)]dxs
dt =0.
The terms in parentheses on the first two lines vanish based on applying (2.45) to the smooth solutions on the sub-intervalsa≤x<xsandxs<x≤brespectively. The remaining terms give the so-calledRankine–Hugoniot shock speed relation
dxs
dt =q(p+(xs,t))−q(p−(xs,t))
p+(xs,t)−p−(xs,t) . (2.48) For the inviscid Burgers equation (2.39), the flux isq(p)=12p2, and for our specific example, p−(x,t) = (1+x)/(1+t)andp+(x,t) = 0 yielding the shock speed equation,
dxs
dt = p+(xs,t)−p−(xs,t)
2 =⇒ dxs
dt = 1+xs
2(1+t). (2.49) Initial conditions for this equation are determined by the time and position where characteristics first cross, necessitating the insertion of a shock; in this case, xs(1)=1. Consequently the position of the shock is given by
xs(t)=
2(1+t)−1 for t≥1.
Fig. 2.5 Characteristic curves in thexttruncated by the shockx=xs(t) corresponding to (2.46)
Figure2.5 shows the shock in the xt plane in its role in separating families of characteristics that would otherwise intersect. Excluding the shock curve, Fig.2.5 has a single characteristic curve passing through each(x,t)point and hence describes a single-valued solution of the transport problem. This figure differs from Fig.2.4 (left) only in the wedge-shaped region bounded by the curvesx=1 andx=tthat form the boundaries of what is sometimes called theshock envelope.
Returning to Fig.2.3, we note that our solutionp(x,t)began as an equilateral triangle on−1≤x≤1 with height one, and hence area one. As time increases the profile steepens toward the right while maintaining its base and height, and hence area, even as it transitions from being an acute triangle (single-valued solution) to an obtuse triangle (multivalued solution) (also see Fig.2.4(left)). The consequence of introducing the shock is to cut out the portion of (2.43) that overturns while modifying the domains on which the other parts of the solution apply fort≥1,
p(x,t)=
⎧⎪
⎨
⎪⎩
0 x<−1,
(1+x)/(1+t) −1≤x≤xs(t),
0 x>xs(t).
(2.50)
The resulting solution profiles are right triangles on the base −1 ≤ x ≤ xs(t) andpranging over 0 ≤ p ≤ maxp = p−(xs(t),t) = √
2/(1+t). To satisfy the conservation of the integral ofp, (2.48) ensures that the shock maintains the area of A=12(base)(height)= 12(√
2(1+t)−1−(−1))(1/√
2(1+t))=1. This illustrates why the shock selection rule is sometimes called theequal-area rule.2Later, we will see in Chap.5that solution (2.50) can also be obtained as asimilarity solution.
2See Fig.2.4—the placement of the shock not only conserves the area of the newly-formed right triangle, but also requires that the areas of the two cut-off multi-valued regions from the obtuse triangle to be equal.