1 modeling maps a physical process to a mathematical representation (e.g. equations) that can be...

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Modeling maps a physical process to a mathematical representation (e.g. equations) that can be solved.

Any physical process is infinitely complex(atom is not the end, there are quarks, …)

To be solvable, we have to make approximations.

Stating the problem (what we want), and making good approximations (explicitly declared), so it can be solved

relatively accurately and efficiently, is the art of modeling.

Good model captures the essential features about the particular aspect of the process we would like to know.

→ Modeling & approximations must be goal-oriented.

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r

The simplest material model is that of point mass.(m, x): no size or orientation, just mass & position.

Earth has size.But Issac Newton (1642-1727) said:

consider Earth as a point mass…and Sun as a point mass

(drawn as finite so you can see, but conceptually, Zip, Nada, Nothin…)

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What do we want to know?The orbital period… the length of a year.

Not just Earth, also Mars, Venus… Moon… comets… eclipsesNewton wanted to unravel the astronomical mysteries that took thousands of Sumerian, Egyptian, Maya… priests hundreds of

years to “figure out” in one stroke. Just plug in different parameters (remarkably few) into the same equation.

Such is the power of a physical model.

Basic ingredients:m(d2x/dt2) = F(x) -- main equation

F(x) = -GmM(x/|x|3) -- “constitutive law”G = 6.673 × 10-11 kg-1m3s-2 -- universal constant

M, m -- material parametersx, t: -- variables

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The traffic flow problem

Step 1. Define scope & goals of the model:To understand traffic pattern on one-way road with no entrance/exit:- What is the maximum carrying capacity of the road?- How long does it take to travel between two points on the road?- If there is traffic jam, how does it move, and how long to clear?Step 2. Make conceptual sketch and define basic quantities:

xx3, v3, L3 x4, v4, L4 x5, v5, L5x2, v2, L2x1, v1, L1

aperture of width D andcentered at x

N(D,x,t): number of vehicles in [x-D/2, x+D/2) at time t.(x,t) ≡ N(D, x, t) / D is the densityv(x,t): the average velocity(x,t) and v(x,t) are coarse-grained, continuous fields.

[ , )2 2

( , ) ( ) / ( , , )i

D D ix x xv x t v N D x t

realistic D ~ 600ftif vehicle spacing 30ft

agent-based description coarse-grained, field description

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Step 3. Develop a mathematical description of the conceptual sketch:The molecular dynamics approach:Write down evolution equation for discrete degrees of freedom: {xi(t)}→ vi(t) = f(…, xi-2(t), xi-1(t), xi(t), xi+1(t), xi+2(t), …) for every i→ Update: …, xi(t+t)=xi(t)+vi(t)t, … → {xi(t+t)}Repeat and this forms an autonomous loop (a set of ODEs).In general, one could also put in randomness.

The continuum, or coarse-grained, approach:The degree of freedom (DoF) are continuous fields such as (x,t), v(x,t) Q(x,t) ≡ (x,t)v(x,t) is the vehicle flux

Step 4. Write down equations that govern the variables:

x

realistic x ~ 1000ft

Q(x,t) Q(x+x,t)

x x+x vehicles in – vehicles out =vehicles accumulated

Q(x,t)dt - Q(x+x,t)dt = (x,t+dt)x - (x,t)x

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Step 5. Introduce constitutive relations between the variables:Intuitively we can appreciate that there should be some relationship

between v and . Your driving experience differs greatly in a bumper-to-bumper jam at 5PM versus an empty lane at 3AM!

We make the following general observations about v():1. v(=0) is still finite.2. v( decreases monotonically with increasing .3. There exists a max where v(max) ≈ 0. (1 mile = 5280 feet)

( , ) - ( , ) ( , )( , ) - ( , )

( , ) - ( , ) ( , )

( , ) ( , )

( )0 0

x

x

t x

x t dt x t x tQ x t Q x x t x x

dt t

Q x t Q x x t x t

x t

Q x t x t

x t

Q v

t x t x

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The simplest v() relation that satisfies the above is: v() = vmax(1-/max) Let’s see where this assumption leads us… (modeling…)

Q()=v()=vmax(1-/max)=maxvmax(/max)(1-/max)

0 1/max

v() Q()

vmax

QmaxQmax is only maxvmax/4!

Why when there is a fire, people shouldn’t rush to the exit…max/2 is the critical density above which a stampede could happen.

Step 6. Reduce the governing equations by approximations: None.

“downward spiral” or catastrophe

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Step 7. Non-dimensionalize the equations by scaling the variables:

max max max max

* * **

maxmax

* * ** max

*

* * **

C * *max C

( ( / )(1- / ))0

( (1- ))0

( (1- ))Suppose the road is [0, ] : 0

( (1- ))Define , and 0.

Alternatively, w

v

t x

vt x

x vL x

L t L xL t

t tv t t x

Cmax

* * *max

C C* * *

CC

max C* * *

1e can use another characteristic length .

( (1- ))0

( (1- )), 0.

( (1- ))Indeed, we can take any , there is still 0.

( ) ( )

l

x vx

l t l xl t

t tv t t x

t x

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Step 8. Solve the dimensionless equations to infer the behavior of the system:

* * * * **

* * * *

* ** * * * *

* *

*

( (1- ))0 (1 2 ) 0.

Define ( 1 2 ( ) 0.

Nonlinear partial differential equation (PDE). it is linear PDE, then should also be a solution, for any

t x t x

c ct x

If

* * * *

* **

* *

0

However, let us first ( ) is indepedent of :

0, or more generally, 0,

with the initial condition that ( , 0) ( ).This is a wave equation, the solution i

pretend c cf f

c ct x t x

f x t f x

0s simply ( , ) ( ).f x t f x ct

x

f (x,0)=f0(x) f(x,1) f(x,2)

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c is therefore identified as the wave speed in the linear ODE problem.

What is the significance of the linear ODE solution in the context of the nonlinear ODE problem?

Consider a homogeneous traffic flow: *(x,t)=*0

This is a solution to the nonlinear PDE.

Now consider a small disturbance in traffic flow pattern: *(x,t) = *0 + *(x,t)* *

* ** *

* ** *0 0

0* *

* *0* *

* ** * * * 20

0 0* *

( ) 0.

( ) ( )( ) 0.

( ) ( )( ) 0

( ) ( ) ( )( ) ( ) ( ) ... 0

2

ct x

ct x

ct x

cc c

t x

11Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations (Cambridge University Press, 1996).

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c(*0 ) is therefore the propagation speed of an infinitesimal disturbance on the originally homogeneous flow pattern of *(x,t)=*0.

* * * *0 0* * *

* * 20* *

( ) ( ) ( )( ) ( ) ... 0

( ) ( )( ) (( ) ) 0.

c ct x x

c Ot x

-vmax

Qmax

0 1/max

v() c()

vmax

0 0

disturbancepatternstationary

disturbance patterntravels backward

disturbance patterntravels forward,but with slowerspeed than theaverage vehicle speed.

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Press, Teukolsky, Vetterling, Flannery, Numerical recipes: the art of scientific computing, 3rd ed.

(Cambridge University Press, 2007)

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