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)

ExampleLet’s solve:

FORTRAN code

Multi-Step methods

-1-1

Adams-Bashforth2 steps:

4 steps:

(So far) Explicit methodsFuture = 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:C:\Users\telzur\Documents\BGU\Teaching\ComputationalPhysics\2011A\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);Plot(t1,y1,’r’);hold onplot(t2,y2,’b’);hold off

Demo folder: Users\telzur\Documents\BGU\Teaching\ComputationalPhysics\2011A\Lectures\05\Matlab

Output

http://www.scilab.org

Parallel tools for Multi-Core and Distributed Parallel Computing

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

Xcos demo