Week2 ODEs

8/13/2019 Week2 ODEs

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Friday, March 15, 13


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CLASS II : Ordinary Differential Equations

Example : Newtons Equation of Motion

m d ~ v (t )

dt =

~

F (t )

~ v (t ) = d ~ x ( t )

dt

m : mass, ~ x (t ) : position , ~ v(t ) : velocity, ~

F (t) : Forcem d

2 ~

x (t )dt 2

= ~

F (t )

First or er High r erSingle equation Systems of equations

Linear Non-linearHomogenous Inhomogenous

Initial Value Problem (IVP) Boundary Value Problem (BVP)

CLASSIFICATIONS of ODEs



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CLASS II : Ordinary Differential Equations

Almost all equations of higher order can be reduced to first order equations

d ~ x

dt =

~ u

d ~ u

dt =

~

F

m

d 2~

xdt

=

~

F m

2nd order ODE

System of 1st Order ODEs

More generally: y( n ) = f (x, y, y . . . , y ( n 1) )

we set: y1 = y

y j = y(j 1) , j = 2 , . . . , n 1

y 0j = yj +1 , j = 1 , 2, . . . , n 1y 0n = f (x,y,y 1 , . . . , y n )

To obtain:



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Numerical Differentiation

How to estimate derivatives in the computer?

Estimate derivatives of a function using a finite set of points.

Assume that the data are the exact values of a smooth function at the data points andfurther, that the derivatives are needed only at the data points.

There are 3 ways to do this:1.INTERPOLATE: Then differentiate the interpolant

2.TAYLOR SERIES

: more effort but have error estimates3.INTEGRATE the ODEs using numerical quadrature



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1. INTERPOLATION

INTERPOLATION

Use Lagrange interpolation: (1st order) (linear interpolation)

f (x ) i

x i x i 1f (x i 1 ) +

i 1

x i x i 1f (x i ) on x i 1 x x i

Differentiating this function we nd that its derivative is:

Backward Derivative

Forward Derivative

f 0 (x i 1 ) D + f = f i f i 1

h i

f 0 (x i ) D f = f i f i 1

h iwith f i = f (x i ) & h i = x i x i 1



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INTERPOLATION (Cont.)

Differentiating this expression and evaluating at the central point of this interval x_i, we get for h_i = h_{i-1} = h

f 0 (x i ) f i +1 f i 1

2h =

1

2(D + D )f (central di ff erences)

f 0 (x i 1 ) 3f i 1 + 4 f i f i +1

2h

f 0 (x i +1 ) f i 1 3f i 3f i +1

2h

Higher order formulas can be obtained by using the quadratic Lagrange polynomial

f (x ) = (x x i )(x x i +1 )

(x i 1

x i )(x i 1

x i +1 )f i 1 +

x x i 1 )(x x i +1 )

(x i

x i 1 )(x i

x i +1 )f i +

(x x i 1 )(x x i )

(x i +1

x i 1 )(x i +1

x i )f i +1

We can also differentiate {LagrangePolynomial} to get a second order derivative

f 00 f i 1 2 f i + f i +1

h2

= D + D = D D + f



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2. TAYLOR SERIES

Start from f (x h ) = f (x ) hf 0 (x ) + h 2

2 f 00 (x ) . . .

and form linear combinations of the function at various mesh points

af (x h ) + bf (x ) + cf (x + h)=( a + b + c)f (x ) + ( c a )hf 0 (x )+

(c + a)1

2h 2 f 00 (x ) + ( c a )

1

6h 3 f 000 (x ) + h.o.t.

Since we wish to approximate $f'(x)$ we wish that coefcientsof f' are 1 and as many of the others are equal to 0

a + b + c = 0

c a =1

h

c + a = 0

Higher Accuracy can be obtained with More points. That can be more expensive and may be problematic with boundary conditions

c = a = 1

2 h and b = 0

Solve to get:

12h

f (x h ) + 12h

f (x + h)= f 0 (x ) + 16

h 2 f 000 (x ) + h.o.t.SUBSTITUTE

EXAMPLE: construct an approximation for the rst derivative of the function at x using values at f(x-h), f(x), f(x+h).

x, x , x 2



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3. INTEGRATE : Numerical Quadrature

dy

dx = f (x, y ) y(x ) = y(0) +

Z

x

0

f (x 0 , y ) dx 0

We can apply any type of quadrature rule for computing

Suppose at x = x n we have y (x ) = y (x n ) = y n .How to compute y (x n +1 ) = y n +1 at x n +1 ?

y(x n +1 ) = y(x n ) + Z x n +1

x n

f (x 0 , y ) dx 0

Forward: Z x n +1

x n

f (x, y ) dx hf (x n , y n ) yn +1 = yn + hf (x n , y n ) (Explicit Euler)

Backward: Z x n +1

x n

f (x, y ) dx hf (x n +1 , y n +1 ) yn +1 = yn + hf (x n +1 , y n +1 ) (Implicit Euler)

Midpoint: Z x n +1

x n

f (x, y ) dx hf (x n + 12

, y n + 12

) yn +1 = yn 1 + 2hf (x n , y n ) (Leap-frog)

Trapezoidal: Z x n +1

x n

f (x, y ) dx h2

[f (x n , y n ) + f (x n +1 , y n +1 )] yn +1 = yn + h2

[f (x n , y n ) + f (x n +1 , y n



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STABILITY OF ODEs

Use of a model equation to study the stability behavior of various numerical schemes

dx

dt = x

Note that this model can be related to any generic right hand side as:

f (x, t ) = f (x 0 , t 0 ) + ( x x 0 ) f x

+ ( t t 0 ) f t

+ h.o.t.

So we write

Stability is determined by the term

dxdt

= f (x, t ) x + ( 0 + 1 t )

Return to Model Equation the solution is: x (t ) = x 0 e t



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STABILITY OF NUMERICAL METHODS

Forward Euler:

Note that :

Euler's method reproduces the rst two terms so it is rst order accurate

xn +1 = x n + h x n x n +1 = x n (1 + h )

x n = x 0 (1 + h ) n

e h = 1 + h + 12

( h )2 + . . .

What if then = k + i e h = ekh (cos h + i sin h )



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BREAK



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d~

x idt =

~

u i , i = 1 , . . . , N

md

~

u idt

= ~

F i , i = 1 , . . . , N

N-body Problems

Particles moving according to Newtons Laws of Motion:

d ~ x

dt =

~ u

m

d ~ u

dt =

~

F

One Particle:

N Particles:(N-body Problem)

~

F i = ~

F i (m 1 , . . . , m N , ~ x 1 , . . . , ~ x N )



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Time Integrators for ODEs in Particle Methods

The beating heart of any Particle simulation :

Replace Ordinary Differential Equations with Difference Equations

Approximate solution snapshots at discrete steps

QUESTIONS

What properties of the continuous system must be preserved in the discrete ?

How accurate ? How to design schemes that respect physical principles regardless of classical

accuracy requirements ?



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Time integration schemes for particle methods

Particle ODEs

Example: For a particle with charge q moving in an electric field Eand magnetic field B, F is the sum of the electrical force qE andthe magnetic force (Lorentz force) qu x B).

Each particle is described by a set (x,u,m).



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Example : The Leapfrog scheme

The Leapfrog Scheme:

In general

If bk is zero the scheme is explicit , else it is implicit andsolutions are found iteratively .



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Criteria to select a scheme:

Consistency,

Accuracy,

Stability ,

Efficiency

Important notions for Certain Physical Systems:Preservation of time symmetriesConservation of physical quantities such as E, mu.



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CONSISTENCY

is consistent with

Why ?

Eulerscheme

The approximate scheme is consistent with the

systems it approximates

EXAMPLE

Taylor series expansion:



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ACCURACY

Accuracy : The deviation of the computed values of position andvelocities from what the code gives are small.

Accuracy is concerned with local errors, i.e. with truncation errorscaused by representing continuous variables by a discrete set ofpoints.



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STABILITY and Eigenvalues

Understand stability through an example ofthe harmonic oscillator

Asymptotically StableA numerical method is asymptotically stable if the growth

of the solution for a linear model problem is asymptoticallybounded

ConditionsEigenvalues (of the linear model problem) must be on orinside the unit disk of the complex plane and not repeatedif on the unit circle

Examples of Numerical Methods: Explicit Euler

Strmer-VerletFriday, March 15, 13


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Time symmetry

Example:The leapfrog scheme is time centered, since

A desired property for particle integrators is that they reflect

time symmetry.

The particle equations are time reversible . Time reversibleapproximations are obtained by defining time-centered

derivatives.

is centered around

is centered around



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and the right hand side is centered around

It is not always practical to employ properly time-centeredschemes as generally they lead to implicit equations in variablesat the new time level.

The Euler scheme is not time centered as

is centered around



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Time Reversible schemes

Consider:

Time reversibility ! Often implicit schemes!



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Conservative Systems

If ~

F i = ~ x i V (~

x 1 , . . . ,~

x N )

~

F i is a conservative force

Conservative System: A system with an energy functionthat is constant along a trajectory(in the phase space )( ~ x, ~ u )

Examples: (1) Gravitation (2) Molecular Dynamics (3) Wave Equation



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Solving Mechanical Systems

M d ~ x

dt 2 = V

Rewrite using velocity: and~ u =d ~ x

dt M

d ~ u

dt = V

- Use different approximations (time integrators) fordifferent components of the solution

- Useful when there is a dichotomy between velocitiesand positions

Examples:(1) Partitioned Runge Kutta Methods(2) Asymmetric Euler A(3) Asymmetric Euler B(4) Strmer-Verlet Methods and Alternatives



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Time Integrators : The Verlet Algorithm

The Verlet algorithm (1967)Use positions r(t), accelerations a(t) and r(t-dt), velocities do notappear.

r(t+dt) = r(t) + dt*u(t) + dt 2*a(t)/2 + dt 3*a(t) + O(dt 4)

r(t-dt) = r(t) dt*u(t) + dt2

*a(t)/2 - dt3

*a(t) + O(dt4

) Add these two expressions to get :

r(t+dt) = 2*r(t) - r(t-dt) + a(t)*dt 2 + O(dt 4)

This is equivalent to central differences of the 2nd derivative We need the velocities to compute the kinetic energy:

u(t) = (r(t+dt) - r(t-dt))/2dt + O (dt^2)



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The Verlet Algorithm

Note that u(t) can be computed only after r(t+dt) is available, which is somewhat awkward.

It is properly centered, r(t+dt) and r(t-dt) play symmetricalroles which makes it time reversible.

Advancement of positions take place in one go as opposedto the two-stages in the Predictor-corrector algorithm.

It is guaranteed to conserve linear momentum and energy.

Also note that in r(t+dt) and r(t-dt) a small term dt2

a(t) isadded to a difference of large terms O(dt) in order tointroduce the trajectory.



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Modified Verlet Algorithms

Modification I:

u(t+dt/2) = u(t-dt/2) + dt*a(t)

r(t+dt) = r(t) + dt u(t+dt/2)

Store r(t), a(t), u(t-dt/2)u(t) = ! [u(t+dt/2) + u(t-dt/2) ]

The Energy at time t can be calculated

This scheme is algebraically equivalent to Verlet, thevelocities appear explicitly but not at time (t)

Less roundoff error problems



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R K M h d


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Runge Kutta Methods

(1) Higher order methods(2) Do not require knowing past values of the solution (Still one step methods)(3) Evaluate s stages

~ x k +1 = ~ x k + ts

Xi =1

bi ~ f (~ x i )

~ x i = ~ x k + tsX

i =1

ij ~ f (~ x j ) i = 1 , . . . s


Partitioned Runge Kutta Methods


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use RK with { ij , bi } to discretize

Partitioned Runge Kutta Methods

Runge Kutta Nystrm Methods

d~

xdt =

~

g(~

x,~

u )d

~

udt

= ~

h ( ~ x, ~ u )

use RK with { ij , bi } to discretize


S V l M h d


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Strmer-Verlet Method

Step 1: M ~ u n + 12 = M ~ u n t

2 V (~ x n )

M ~ u n +1 = M ~ u n + 12 t

2 V (~ x n +1 )

~ x n +1 = ~ x n + t ~ u n + 12Step 2:

Step 3:

Alternative 1: solve the velocity at half steps onlyAlternative 2: eliminate the velocity


P i l M d l


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PhysicalSystem Model(Differential Equations) Simulation(Difference Equations)

want to maintain geometricproperties of physical system(i.e conservative systems)

Particle Methods (N body problem)Solve using time integratorsODEs = DEs(accuracy and stability )

Particle Models

Week2 ODEs

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