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8/13/2019 hyperbolic systems of conservation law
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The Scalar Conservation Law
ut+ f(u)x= 0 u: conserved quantity f(u) : flux
d
dt b
a
u(t, x) dx = b
a
ut(t, x) dx = b
a
f(u(t, x))x dx
= f(u(t, a)) f(u(t, b)) = [inflow at a] [outflow at b] .
a b x
u
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Example : Traffic Flow
x
a b
= density of cars
ddt
ba
(t, x) dx= [flux of cars entering at a] [flux of cars exiting at b]
flux: f(t, x) =[number of cars crossing the point x per unit time]
= [density][velocity]
t +
x[ v()] = 0
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Weak solutions
a b x
u
conservation equation: ut+ f(u)x = 0
quasilinear form: ut+ a(u)ux = 0 a(u) =f(u)
Conservation equation remains meaningful for u= u(t, x)
discontinuous, in distributional sense: {ut+ f(u)x} dxdt = 0 for all C
1c
Need only : u, f(u) locally integrable
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Convergence
ut+ f(u)x = 0
Assume: un is a solution for n 1,
un u f(un) f(u) in L
1
loc
then
{ut+ f(u)x} dxdt = limn
{unt+ f(un)x} dxdt = 0
for all C1c . Hence u is a weak solution as well.
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Systems of Conservation Laws
tu1+
xf1(u1, . . . , un) = 0,
t un+
x fn(u1, . . . , un) = 0.
ut+ f(u)x= 0
u= (u1, . . . , un) IRn conserved quantities
f = (f1, . . . , f n) :IRn IRn fluxes
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Euler equations of gas dynamics (1755)
t+ (v)x = 0 (conservation of mass)
(v)t+ (v2 +p)x = 0 (conservation of momentum)
(E)t+ (Ev +pv)x = 0 (conservation of energy)
= mass density v = velocity
E=e + v2/2 = energy density per unit mass (internal + kinetic)
p= p(, e) constitutive relation
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Scalar Equation with Linear Flux
ut+ f(u)x = 0 f(u) =u + c
ut+ ux = 0 u(0, x) =(x)
Explicit solution: u(t, x) =(x t)
traveling wave with speed f(u) =
u(t)
x
t
u(0)
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A Linear Hyperbolic System
ut+ Aux= 0 u(0, x) =(x)
1 < < n eigenvalues
r1, . . . , rn right eigenvectors
l1, . . . , rn left eigenvectors
Explicit solution: linear superposition of travelling waves
u(t, x) =
ii(x it)ri i(s) =li (s)
x
2u
1u
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Nonlinear Effects
ut+ A(u)ux = 0
eigenvalues depend on u = waves change shape
x
u(0)u(t)
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eigenvectors depend on u = nontrivial wave interactions
tt
x x
linear nonlinear
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Loss of Regularity
ut+ f(u)x= 0 u(0, x) =(x)
Method of characteristics yields: u
t, x0+ t f((x0))
= (x0)
characteristic speed = f(u)
x
u(0) u(t)
Global solutions only in a space of discontinuous functions
u(t, ) BV
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Wave Interactions
ut = A(u)ux
i(u) =i-th eigenvalue li(u), ri(u) =i-th eigenvectors
uix.
= li ux = [i-th component of ux] = [density of i-waves in u]
ux =n
i=1
uixri(u) ut = n
i=1
i(u)uixri(u)
differentiate first equation w.r.t. t, second one w.r.t. x
= evolution equation for scalar components uix
(uix)t+ (iuix)x =
j>k
(j k)
li [rj, rk]
ujxukx
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source terms: (j k)
li [rj, rk]
ujxukx
= amount of i-waves produced by the interaction of j-waves with k-waves
j k = [difference in speed]= [rate at which j-waves and k-waves cross each other]
ujxukx = [density of j-waves] [density of k-waves]
[rj, rk] = (Drk)rj (Drj)rk (Lie bracket)
= [directional derivative of rk in the direction of rj][directional derivative of rj in the direction of rk]
li [rj, rk] = i-th component of the Lie bracket [rj, rk] along the basisof eigenvectors {r1, . . . , rn}
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Shock solutions
ut+ f(u)x = 0
u(t, x) = u if x < tu
+
if x > t
is a weak solution if and only if
[u+ u] = f(u+) f(u) Rankine - Hugoniot equations
[speed of the shock] [jump in the state] = [jump in the flux]
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Derivation of the Rankine - Hugoniot Equations
ut+ f(u)x
dxdt = 0 for all C
1c
v .=
u , f (u)
t
x
n
n+
+
=x t
Supp
u = u+
u = u
0 =
+
divv dxdt =
+
n+ v ds +
n v ds
=
[u+ f(u+)] (t,t) dt +
[ u + f(u)] (t,t) dt
=
(u+ u) (f(u+) f(u))
(t,t) dt .
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Alternative formulation:
(u+
u
) = f(u+
) f(u
) = 1
0 Df(u+
+ (1 )u
) (u+
u
) d
= A(u+, u) (u+ u)
A(u, v) .=
1
0
Df(u + (1 )v) d = [averaged Jacobian matrix]
The Rankine-Hugoniot conditions hold if and only if
(u+ u) = A(u+, u) (u+ u)
The jump u+
u
is an eigenvector of the averaged matrix A(u+
, u
) The speed coincides with the corresponding eigenvalue
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scalar conservation law: ut+ f(u)x= 0
u+ u
f(u)
u +
u
x
t
f (u)
= f(u+) f(u)
u+ u =
1
u+ u
u+u
f(s) ds
[speed of the shock] = [slope of secant line through u, u+ on the graph of f]
= [average of the characteristic speeds between u and u+]
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Points of Approximate Jump
The function u = u(t, x) has an approximate jump at a point (, )if there exists states u =u+ and a speed such that, calling
U(t, x) .=
u if x < t,u+ if x > t,
there holds
lim0+
1
2
+
+
u(t, x) U(t , x )
dxdt= 0 (2)
.
x =
x
t
u
u+
Theorem. If u is a weak solution to the system of conservation lawsut +f(u)x= 0 then the Rankine-Hugoniot equations hold at each pointof approximate jump.
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Construction of Shock Curves
Problem: Given u IRn, find the states u+ IRn which, for some
speed , satisfy the Rankine - Hugoniot equations
(u+ u) = f(u+) f(u) = A(u, u+)(u+ u)
Alternative formulation: Fix i {1, . . . , n}. The jump u+ u is a(right) i-eigenvector of the averaged matrix A(u, u+) if and only if itis orthogonal to all (left) eigenvectors lj(u
, u+) ofA(u, u+), for j =i
lj(u, u+) (u+ u) = 0 for all j =i (RHi)
Implicit function theorem = for each i there exists a curve
sSi(s)(u) of points that satisfy (RHi) i
u
Si
r
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Weak solutions can be non-unique
Example: a Cauchy problem for Burgers equation
ut+ (u2/2)x= 0 u(0, x) =
1 if x 0,0 if x
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Admissibility Conditions on Shocks
For physical systems:
a concave entropy should not decrease
For general hyperbolic systems:
require stability w.r.t. small perturbations
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Stability Conditions: the Scalar Case
Perturb the shock with left and right states u, u+ by
inserting an intermediate state u
[u
, u+
]
Initial shock is stable
[speed of jump behind] [speed of jump ahead]
f(u) f(u)
u u
f(u+) f(u)
u+ u
u
u
u
u
_
*
xx
+u
*u
_ +
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speed of a shock = slope of a secant line to the graph of f
f
u uu u + -+-
f
u u* *
Stability conditions:
when u < u+ the graph of f should remain above the secant line
when u > u+, the graph of f should remain below the secant line
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General Stability Conditions
Scalar case: stability holds if and only if
f(u) f(u)
u u
f(u+) f(u)
u+ uf(u)
u
u+u
+u*u u*
f(u)
Vector valued case: u+ =Si()(u) for some IR.
Admissibility Condition (T.P.Liu, 1976). The speed () of theshock joining u with u+ must be less or equal to the speed of everysmaller shock, joining u with an intermediate state u = Si(s)(u),s [0, ].
(u, u+) (u, u)
+u u* u = S ( ) (u )i
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x = (t) x = t / 2
Admissibility Condition (P. Lax, 1957) A shock connecting thestates u, u+, travelling with speed = i(u, u+) is admissible if
i(u) i(u
, u+) i(u+)
[Liu condition] = [Lax condition]
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Entropy Admissibility Condition
A weak solution u of the hyperbolic system ut+ f(u)x= 0
is entropy admissible if
[(u)]t+ [q(u)]x0
in the sense of distributions, for every pair (, q), where is a
convex entropy and q is the corresponding entropy flux.
{(u)t+ q(u)x} dxdt 0 C
1c , 0
- smooth solutions conserve all entropies
- solutions with shocks are admissible if they dissipate all convex en-tropies
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Existence of entropy - entropy flux pairs
D(u) Df(u) =Dq (u)
u1
un
f1u1
f1un
fnu1
fnun
=
qu1
qun
- a system of n equations for 2 unknown functions: (u), q(u)
- over-determined if n >2
- however, some physical systems (described by several conservationlaws) are endowed with natural entropies
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