Dr. Guy Tel-Zur

Computational PhysicsDifferential Equations

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Agenda

• MHJ Chapter 13 & Koonin Chapter 2• How to solve ODE using Matlab• Scilab

Topics

• Defining the scope of the discussion• Simple methods• Multi-Step methods• Runge-Kutta• Case Studies - Pendulum

The scope of the discussion

For a higher order ODE a set of coupled 1st order equations:

Simple methodsEuler method:

Integration using higher order accuracy:

Taylor series expansion:

Local error!

Better than Euler’s method but useful only when it is easy to differentiate f(x,y)

An ExampleLet’s solve:

Boundary conditiion

FORTRAN code - I

Demo: virtual box, folder: ~/fortran, program: chap2a.fCompilation: telzur@linux1:~/fortran$ fort77 -o chap2a chap2a.f

Euler’s method

FORTRAN code - II FUNC(X,Y)=-X*YC-------scientific computing course, lecture 05 - differential equationsC Guy Tel-Zur, April 2011C example chap2a from KooninC compile under ubuntu using: fort77 -o chap2a_taylor chap2a_taylor.f 20 PRINT *,'ENTER STEP SIZE' READ *,h IF (H.LE.0) STOP NSTEP=3./h Y=1. DO 10 IX=0,NSTEP-1 X=IX*H Y=Y+H*FUNC(X,Y)+0.5*H*H*(-Y-FUNC(X,Y)*X) DIFF=EXP(-0.5*(X+H)**2)-Y PRINT *,IX,X+H,Y,DIFF 10 CONTINUE GOTO 20 END

Taylor’s series method

The Results

IX X Y DIFF Y DIFF 0 0.5 1. -.117503099 .875 .00749690272 1 1.0 .75 -.143469334 .57421875 .0323119089 2 1.5 .375 -.0503475331 .287109375 .0375430919 3 2.0 .09375 .0415852815 .116638184 .0186970998 4 2.5 0. .0439369343 .0437393188 .000197614776 5 3.0 0. .0111089963 .0177690983 -.00666010194

Euler’s Taylor’s

Multi-Step methodsAdams-Bashforth

2 steps:

4 steps:

(So far) Explicit methods

Future = Function(Present && Past)

Implicit methods

Future = Function(Future && Present && Past)

Let’s calculate dy/dx at a mid way between lattice points:

Rearrange:

This is a recursion relation!

Let’s replace:

A simplification occurs if f(x,y)=y*g(x), then the recursion equation becomes:

An example, suppose g(x)=-x

yn+1=(1-xnh/2)/(1+xn+1h/2)yn

This can be easily calculated, for example:

Calculate y(x=1) for h=0.5

X0=0, y(0)=1X1=0.5, y(0.5)=?x2=1.0, y(1.0)=?

Error=-0.01569

The solution:

Predictor-Corrector method

Predictor-Corrector method

Runge-Kutta

Proceed to:Physics examples: Ideal harmonic oscillator – section 13.6.1

Physics Project – The pendulum, 13.7

I use a modified the C++ code from:http://www.fys.uio.no/compphys/cp/programs/FYS3150/chapter13/cpp/program2.cpp

fout.close fout.close()

Demo on folder:\Lectures\05\CPP

ODEs in Matlabfunction dydt = odefun(t,y)a=0.001;b=1.0;dydt =b*t*sin(t)+a*t*t;

Usage:

[t1, y1]=ode23(@odefun,[0 100],0);[t2, y2]=ode45(@odefun,[0 100],0);plot(t1,y1,’r’);hold onplot(t2,y2,’b’);hold off

Demo folder: C:\Users\telzur\Documents\Weizmann\ScientificComputing\SC2011B\Lectures\05\Matlab

Output

http://www.scilab.org

Parallel tools for Multi-Core and Distributed Parallel Computing

In preparation

Parallel executionA new function (parallel_run) allows parallel computations and leverages multicore architectures and their capacities.

In future Scilab versions:http://help.scilab.org/docs/5.3.1/en_US/parallel_run.html

Xcos demo