A Very Brief Introduction to Conservation Laws Wen Shen Department - - PowerPoint PPT Presentation

A Very Brief Introduction to Conservation Laws

Wen Shen

Department of Mathematics, Penn State University

Summer REU Tutorial, May 2013

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 1 / 28

The derivation of conservation laws

A conservation law is an PDE that describes time evolution of some quantity (quantities) that is (are) conserved in time. Let u(t, x) be the unknown. We have the initial value problem ∂ ∂t u(t, x) + ∂ ∂x f (u(t, x)) = 0, u(0, x) = ¯ u(x). If u(t, x) is a scalar function, then f (·) is a scalar-valued function. Then, the equation is called a scalar conservation law. If u(t, x) ∈ Rn is a vector of length n, the f (·) is a vector-valued function. Then, this is a system of conservation laws. The function f (u) is called the flux. Typically it is a non-linear function.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 2 / 28

A conservation law describing traffic flow

• x

a b

= density of cars b

a

ρ(t, x) dx = total number of cars at time t within the interval [a, b] d dt b

a

ρ(t, x) dx = [flux of cars entering at a] − [flux of cars exiting at b] = f (t, a) − f (t, b)

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 3 / 28

flux: = [number of cars crossing the point x per unit time] = [density]×[velocity] = ρ(t, x) · v(t, x) Assume: v = v(ρ), i.e., speed depends on the density ρ. For example: v(ρ) = k(M − ρ), 0 ≤ ρ ≤ M M=max car density kM=max car speed when ρ = 0. For simplicity, we choose k = 1, M = 1 and get v(ρ) = 1 − ρ. Flux: f (ρ) = ρ · v(ρ) = ρ(1 − ρ). d dt b

a

ρ(t, x) dx = f (ρ(t, a)) − f (ρ(t, b)). This is an integral form of the conservation law.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 4 / 28

d dt b

a

ρ(t, x) dx = f (ρ(t, a)) − f (ρ(t, b)) Integrate in time from t1 to t2: b

a

ρ(t2, x) dx − b

a

ρ(t1, x) dx = t2

t1

f (ρ(t, a)) dt − t2

t1

f (ρ(t, b)) dt,

• r equivalently

b

a

ρ(t2, x) dx = b

a

ρ(t1, x) dx + t2

t1

f (ρ(t, a)) dt − t2

t1

f (ρ(t, b)) dt. This gives an expression for the mass in [a, b] at t2 in terms of the mass at an earlier time t1 and the total integrated flux along the boundary.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 5 / 28

Differential form of the conservation law

Integral form b

a

• ρ(t2, x) − ρ(t1, x)
• dt =

t2

t1

• f (ρ(t, a)) − f (ρ(t, b))
• dt,

Assume that ρ(x, t) is a differentiable function in both x and t, so ρ(t2, x) − ρ(t1, x) = t2

t1

∂ ∂t ρ(t, x) dt f (ρ(t, b)) − f (ρ(t, a)) = b

a

∂ ∂x f (ρ(t, x)) dx Then t2

t1

b

a

∂ ∂t ρ(t, x) + ∂ ∂x f (ρ(t, x))

• dx dt = 0.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 6 / 28

t2

t1

b

a

∂ ∂t ρ(t, x) + ∂ ∂x f (ρ(t, x))

• dx dt = 0.

This holds for all a, b and t1, t2! The integrand must be 0! ∂ ∂t ρ(t, x) + ∂ ∂x f (ρ(t, x)) = 0,

• r with simplified notation

ρt + f (ρ)x = 0. Traffic flow model: ρt + (ρ(1 − ρ))x = 0, ρ(0, x) = ρo(x).

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 7 / 28

System of conservation laws: gas dynamics

The most celebrated example for a system of conservation laws comes from gas dynamics, with the famous Euler equations. Variables: ρ=density, v=velocity, ρ · v=momentum, E=energy, p=the pressure. Three conserved quantities: conservation of mass, momentum and energy. These exactly give us the following 3 × 3 system ρt + (ρv)x = 0, conservation of mass (ρv)t + (ρv 2 + p)x = 0, conservation of momentum Et + (v(E + p))x = 0, conservation of energy. There is an additional equation, where the pressure p is given as a function of

• ther quantities. This is called the “equation of the state”. For example,

p = p(ρ, v).

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 8 / 28

Simplest Case: Linear Transport equation

Consider ut + f (u)x = 0, u(0, x) = ¯ u(x). If f (u) = au + b (linear function), then ut + a · ux = 0, u(0, x) = u0(x). This is the transport equation. Explicit solution u(t, x) = u0(x − at). One can easily verify this but plug it into the equation, and also check the initial condition. The solution is simply the initial profile u0(x) traveling with constant velocity a.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 9 / 28

Method of characteristics: nonlinear conservation law

Consider nonlinear flux f ∈ C2 for ut + f (u)x = 0, → ut + f ′(u)ux = 0. A characteristic is a line t → x(t) such that x′(t) = f ′(u(t, x(t))). The evolution of u along a characteristic: d dt u(t, x(t)) = ut + uxx′(t) = −f ′(u)ux + uxf ′(u) = 0. ⇒ u is constant along a characteristic! ⇒ x′(t) =constant along a characteristic! ⇒ all characteristics are straight lines, with slope=f ′(u(0, x))!

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 10 / 28

Traffic Flow: Speed of cars and characteristic speed

ρt + f (ρ)x = 0, f (ρ) = ρ v(ρ) = ρ(1 − ρ), ρ(0, x) = ¯ ρ(x) v(ρ) = 1 − ρ = speed of cars, depending on the density characteristic speed = f ′(ρ) = 1 − 2ρ ≤ v(ρ) characteristic speed is not the same as the car speed! Characteristics t → x(t) are lines where information is carried along! ρ(t, x(t)) = ¯ ρ(x(0)) = constant

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 11 / 28

Characteristics and particle trajectories

p(t) (x)

• _

x(t) x x t

• (, )

x

• Wen Shen (Penn State)

Conservation Laws Summer REU Tutorial, May 2013 12 / 28

Loss of singularity for nonlinear equations

• x

= constant

• t

(, ) x ( ) t,x (0,x)

Points on the graph of ρ(t, ·) move horizontally, with characteristic speed f ′(ρ). At a finite time τ the tangent becomes vertical and a discontinuity form.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 13 / 28

Formation of Shock waves in finite time

For general scalar conservation law ut + f (u)x = 0, u(0, x) = ¯ u(x) if f is a nonlinear function, i.e., f ′(u) is not constant, then, characteristics initiated at different point of x at t = 0 will have different slope, and they will interact in finite time. → discontinuities will form in finite time even with smooth initial data! → These are called shock waves or shocks! → We must only require u(t, x) bounded and measurable.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 14 / 28

Weak solutions

Discontinuous solutions will not satisfy the differential equation ut + f (u)x = 0 Re-define solution concept, using integral form. b

a

• u(t2, x) − u(t1, x)
• dt =

t2

t1

• f (u(t, a)) − f (u(t, b))
• dt,

u(t, x) is a weak solution if the integral form holds for any a, b, t1, t2. An alternative definition: u = u(t, x) is a weak solution if {uφt + f (u)φx} dx dt = 0 for every positive test function φ ∈ C 1 with compact support.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 15 / 28

Shock propagation; Riemann Problem

Riemann problem: ut + f (u)x = 0, u(0, x) =

• ul,

x ≤ 0 ur, x > 0 Shock speed: let s be the shock speed, and let M > st. u(t, x) =

• ul,

x < st ur, x > st , M

−M

u(x, t) dx = (M + st)ul + (M − st)ur d dt M

−M

u(x, t) dx = sul − sur = s(ul − ur) (by conservation law:) d dt M

−M

u(x, t) dx = f (ul) − f (ur). Rankine-Hugoniot jump condition: s(ul − ur) = f (ul) − f (ur)

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 16 / 28

Manipulating conservation laws

Burgers’ equation: ut + (1 2u2)x = 0, f (u) = 1 2u2, f ′(u) = u, f ′′(u) = 1 > 0 (1) Shock speed: s1 = f (ul) − f (ur) ul − ur = 1 2 (ul)2 − (ur)2 ul − ur = 1 2(ul + ur). Multiply Burgers equation by 2u, 2u · ut + 2u · uux = 0, (u2)t + (2 3u3)x = 0, (2) Shock speed s2 = 2 3 (ul)3 − (ur)3 (ul)2 − (ur)2

• ,

s1 = s2. (1) and (2) are equivalent for smooth solutions, but very different for discontinuous solutions!

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 17 / 28

The equal area rule

Method of characteristics leads to multi-valued functions, after finite time. To get back to single-valued functions, we inserting a shock. The exact location

• f the shock follows the “equal area rule”.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 18 / 28

Observation: Following the equal area rule, the characteristics enter the shock both from the left and from the right. Burgers’ equation:

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 19 / 28

Admissible conditions: Lax’s entropy condition

x(t): location of a shock ul = u(t, x−), ur = u(t, x+): left and right state of the shock. Lax’s entropy condition: f ′(ul) > x′(t) > f ′(ur).

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 20 / 28

Admissible shocks for Burgers equation

f (u) = u2/2, f ′(u) = u, f ′′(u) = 1 > 0 Lax’s condition f ′(ul) > f ′(ur) implies ul > ur. Conclusion: Only downward jumps are admissible. In general, if f ′′(u) > 0 (convex), then f ′(u) is increasing, then only downward jumps are admissible.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 21 / 28

Admissible shocks for traffic flow

f (ρ) = ρ(1 − ρ), f ′(ρ) = 1 − 2ρ, f ′′(ρ) = −2 < 0. Lax’s condition f ′(ρl) > f ′(ρr) implies ρl < ρr. Only upward jumps are admissible. In fact, image cars lined up in front a red light (discontinuous initial data with a downward jump). At t = 0, the light turn green. We never observe this jump in the car density moves forward. Actually, we observe that the cars spread out. In general, for ut + f (u)x = 0, if f ′′(u) < 0 (concave), then f ′(u) is decreasing, so upward jumps are admissible.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 22 / 28

An entropy violating shock: characteristics

Burgers’ equation:

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 23 / 28

Rarefaction waves

convex flux: ut + f (u)x = 0, u(0, x) =

• ul,

x ≤ 0 ur, x > 0, ul < ur. Self similar solution: u depends only on ξ = x/t. Rarefaction fan. u(t, x) =      ul, (x/t) < f ′(ul) ur, (x/t) > f ′(ur) φ(x/t), f ′(ul) ≤ (x/t) ≤ f ′(ur) φ(ξ) : smooth function

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 24 / 28

Case study: traffic jam

ρt + f (ρ)x = 0, f (ρ) = ρ(1 − ρ) Riemann problem: ρ(0, x) =

• 0.5,

x < 0 1, x > 0 Shock speed: s = f (1) − f (0.5) 1 − 0.5 = 0 − 0.25 0.5 = −0.5.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 25 / 28

Numerical methods

Lax-Friedrich Methods: Finite Difference Method. Godunov Methods: Finite Volume Method. ENO and WENO Schemes, and various other higher order methods. Front Tracking Methods: Use piecewise constant approximation. Treat all fronts (jumps) as shock or small rarefactions. Track all fronts. Fronts may interact or

• merge. Solve a new Riemann problem.

Challenge:

Accurately approximate shocks as well as the smooth part of the solution.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 26 / 28

Front tracking: ut +

• u2(1 − u2)
• x = 0,

u(0, x) = sin(πx)

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 27 / 28

References

Randall LeVeque. Numerical Methods for Conservation Laws, Lectures in Mathematics, ETH-Zurich Birkhauser-Verlag, Basel, 1990. ISBN 3-7643-2464-3 Helge Holden and Nils Henrik Risebro. Front Tracking for Hyperbolic Conservation Laws. Springer, (2007). Applied Mathematical Sciences, vol

• 152. ISBN 978-3-540-43289-0.

Randall LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002. ISBN 0-521-00924-3.

Wen Shen (Penn State) Conservation Laws Summer REU Tutorial, May 2013 28 / 28