Computational Physics Differential Equations

1. Computational PhysicsDifferential Equations Dr. Guy Tel-Zur Autumn Colors, by Bobby Mikul, http://www.publicdomainpictures.net Version 10-11-2010 18:30

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

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

4. The scope of the discussion For a higher order ODE  a set of coupled 1st order equations:

5. Simple methods Euler method: Integration using higher order accuracy: Taylor series expansion:

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

7. Example Let’s solve:

9. FORTRAN code

10. Multi-Step methods -1 -1

11. Adams-Bashforth 2 steps: 4 steps:

12. (So far) Explicit methods Future = Function(Present && Past) Implicit methods Future = Function(Future && Present && Past)

13. Let’s calculate dy/dx at a mid way between lattice points: Let’s replace: Rearrange: This is a recursion relation!

14. 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)=1 X1=0.5, y(0.5)=? x2=1.0, y(1.0)=?

15. The solution: Error=-0.01569

16. Predictor-Corrector method

17. Predictor-Corrector method

18. Runge-Kutta

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

22. 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

23. ODEs in Matlab function 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 on plot(t2,y2,’b’); hold off Demo folder: Users\telzur\Documents\BGU\Teaching\ComputationalPhysics\2011A\Lectures\05\Matlab

24. Output

25. http://www.scilab.org

30. Parallel tools for Multi-Core and Distributed Parallel Computing

31. In preparation Parallel execution A new function (parallel_run) allows parallel computations and leverages multicore architectures and their capacities.

37. Xcos demo