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 Summer REU Tutorial, May 2013 1 / Wen Shen (Penn State) Conservation Laws 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 ) ∈ R n 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. Summer REU Tutorial, May 2013 2 / Wen Shen (Penn State) Conservation Laws 28

A conservation law describing traffic flow = density of cars � x a b � b ρ ( t , x ) dx = total number of cars at time t within the interval [ a , b ] a � b d ρ ( t , x ) dx = [flux of cars entering at a ] − [flux of cars exiting at b ] dt a = f ( t , a ) − f ( t , b ) Summer REU Tutorial, May 2013 3 / Wen Shen (Penn State) Conservation Laws 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 ). � b d ρ ( t , x ) dx = f ( ρ ( t , a )) − f ( ρ ( t , b )) . dt a This is an integral form of the conservation law. Summer REU Tutorial, May 2013 4 / Wen Shen (Penn State) Conservation Laws 28

� b d ρ ( t , x ) dx = f ( ρ ( t , a )) − f ( ρ ( t , b )) dt a Integrate in time from t 1 to t 2 : � b � b � t 2 � t 2 ρ ( t 2 , x ) dx − ρ ( t 1 , x ) dx = f ( ρ ( t , a )) dt − f ( ρ ( t , b )) dt , a a t 1 t 1 or equivalently � b � b � t 2 � t 2 ρ ( t 2 , x ) dx = ρ ( t 1 , x ) dx + f ( ρ ( t , a )) dt − f ( ρ ( t , b )) dt . a a t 1 t 1 This gives an expression for the mass in [ a , b ] at t 2 in terms of the mass at an earlier time t 1 and the total integrated flux along the boundary. Summer REU Tutorial, May 2013 5 / Wen Shen (Penn State) Conservation Laws 28

Differential form of the conservation law Integral form � b � t 2 � � � � ρ ( t 2 , x ) − ρ ( t 1 , x ) dt = f ( ρ ( t , a )) − f ( ρ ( t , b )) dt , a t 1 Assume that ρ ( x , t ) is a differentiable function in both x and t , so � t 2 ∂ ρ ( t 2 , x ) − ρ ( t 1 , x ) = ∂ t ρ ( t , x ) dt t 1 � b ∂ f ( ρ ( t , b )) − f ( ρ ( t , a )) = ∂ x f ( ρ ( t , x )) dx a Then � ∂ � t 2 � b � ∂ t ρ ( t , x ) + ∂ ∂ x f ( ρ ( t , x )) dx dt = 0 . t 1 a Summer REU Tutorial, May 2013 6 / Wen Shen (Penn State) Conservation Laws 28

� ∂ � t 2 � b � ∂ t ρ ( t , x ) + ∂ ∂ x f ( ρ ( t , x )) dx dt = 0 . t 1 a This holds for all a , b and t 1 , t 2 ! The integrand must be 0! ∂ t ρ ( t , x ) + ∂ ∂ ∂ x f ( ρ ( t , x )) = 0 , or with simplified notation ρ t + f ( ρ ) x = 0 . Traffic flow model: ρ t + ( ρ (1 − ρ )) x = 0 , ρ (0 , x ) = ρ o ( x ) . Summer REU Tutorial, May 2013 7 / Wen Shen (Penn State) Conservation Laws 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 ρ t + ( ρ v ) x = 0 , conservation of mass ( ρ v ) t + ( ρ v 2 + p ) x = 0 , conservation of momentum E t + ( v ( E + p )) x = 0 , conservation of energy. There is an additional equation, where the pressure p is given as a function of other quantities. This is called the “equation of the state”. For example, p = p ( ρ, v ). Summer REU Tutorial, May 2013 8 / Wen Shen (Penn State) Conservation Laws 28

Simplest Case: Linear Transport equation Consider u t + f ( u ) x = 0 , u (0 , x ) = ¯ u ( x ) . If f ( u ) = au + b (linear function), then u t + a · u x = 0 , u (0 , x ) = u 0 ( x ) . This is the transport equation . Explicit solution u ( t , x ) = u 0 ( 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 u 0 ( x ) traveling with constant velocity a . Summer REU Tutorial, May 2013 9 / Wen Shen (Penn State) Conservation Laws 28

Method of characteristics: nonlinear conservation law Consider nonlinear flux f ∈ C 2 for u t + f ( u ) x = 0 , → u t + f ′ ( u ) u x = 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 )) = u t + u x x ′ ( t ) = − f ′ ( u ) u x + u x f ′ ( 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 ))! Summer REU Tutorial, May 2013 10 / Wen Shen (Penn State) Conservation Laws 28

Traffic Flow: Speed of cars and characteristic speed ρ t + 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 − 2 ρ ≤ 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 Summer REU Tutorial, May 2013 11 / Wen Shen (Penn State) Conservation Laws 28

Characteristics and particle trajectories � x � ( � , ) t � x(t) p(t) _ � (x) 0 x x 0 Summer REU Tutorial, May 2013 12 / Wen Shen (Penn State) Conservation Laws 28

Loss of singularity for nonlinear equations t � � = constant 0 x � (0,x) x � ( ) t,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. Summer REU Tutorial, May 2013 13 / Wen Shen (Penn State) Conservation Laws 28

Formation of Shock waves in finite time For general scalar conservation law u t + 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. Summer REU Tutorial, May 2013 14 / Wen Shen (Penn State) Conservation Laws 28

Weak solutions Discontinuous solutions will not satisfy the differential equation u t + f ( u ) x = 0 Re-define solution concept, using integral form. � b � t 2 � � � � u ( t 2 , x ) − u ( t 1 , x ) dt = f ( u ( t , a )) − f ( u ( t , b )) dt , a t 1 u ( t , x ) is a weak solution if the integral form holds for any a , b , t 1 , t 2 . An alternative definition: u = u ( t , x ) is a weak solution if � � { u φ t + f ( u ) φ x } dx dt = 0 for every positive test function φ ∈ C 1 with compact support. Summer REU Tutorial, May 2013 15 / Wen Shen (Penn State) Conservation Laws 28

Shock propagation; Riemann Problem � u l , x ≤ 0 Riemann problem: u t + f ( u ) x = 0 , u (0 , x ) = u r , x > 0 Shock speed: let s be the shock speed, and let M > st . � M � u l , x < st u ( x , t ) dx = ( M + st ) u l + ( M − st ) u r u ( t , x ) = x > st , u r , − M � M d u ( x , t ) dx = su l − su r = s ( u l − u r ) dt − M � M d u ( x , t ) dx = f ( u l ) − f ( u r ) . (by conservation law:) dt − M s ( u l − u r ) = f ( u l ) − f ( u r ) Rankine-Hugoniot jump condition: Summer REU Tutorial, May 2013 16 / Wen Shen (Penn State) Conservation Laws 28

Manipulating conservation laws Burgers’ equation: u t + (1 f ( u ) = 1 2 u 2 ) x = 0 , 2 u 2 , f ′ ( u ) = u , f ′′ ( u ) = 1 > 0 (1) Shock speed: ( u l ) 2 − ( u r ) 2 s 1 = f ( u l ) − f ( u r ) = 1 = 1 2( u l + u r ) . u l − u r u l − u r 2 Multiply Burgers equation by 2 u , ( u 2 ) t + (2 3 u 3 ) x = 0 , 2 u · u t + 2 u · uu x = 0 , (2) Shock speed � ( u l ) 3 − ( u r ) 3 s 2 = 2 � , s 1 � = s 2 . ( u l ) 2 − ( u r ) 2 3 (1) and (2) are equivalent for smooth solutions, but very different for discontinuous solutions! Summer REU Tutorial, May 2013 17 / Wen Shen (Penn State) Conservation Laws 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 of the shock follows the “equal area rule”. Summer REU Tutorial, May 2013 18 / Wen Shen (Penn State) Conservation Laws 28

Observation : Following the equal area rule, the characteristics enter the shock both from the left and from the right. Burgers’ equation: Summer REU Tutorial, May 2013 19 / Wen Shen (Penn State) Conservation Laws 28