hyperbolic systems of conservation law

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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hyperbolic systems of conservation law