1. Computational Physics Differential Equations
Dr. Guy Tel-Zur
Autumn Colors, by Bobby Mikul,
Version :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: Taylor series expansion:
Integration using higher order accuracy:
Taylor series expansion:
Think of ‘x’ as the time ‘t’

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

7. An Example
Boundary conditiion
Let’s solve:

9. FORTRAN code - I Euler’s method
Demo: virtual box, folder: ~/fortran, program: chap2a.f
Compilation: fort77 -o chap2a chap2a.f

10. Taylor’s series method
FORTRAN code - II
Taylor’s series method
FUNC(X,Y)=-X*Y
C scientific computing course, lecture 05 - differential equations
C Guy Tel-Zur, April 2011
C example chap2a from Koonin
C 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

11. The Results Taylor’s Euler’s IX X Y DIFF Y DIFF

12. Multi-Step methods -1 -1
Source:: Koonin page 27

13. Multi-Step methods Adams-Bashforth
2 steps:
4 steps:

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

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

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

17. The solution:
𝑦 1 = 1− 𝑦( 1 2 )
𝑦 = 1−0∙0.5/2 1+1∙0.5/2 ∙1=0.8889
𝑦 1 =
𝑦 1 𝑒𝑥𝑎𝑐𝑡= 𝑒 −1/2 =
Error=

18. Predictor-Corrector method

19. Predictor-Corrector method

20. Runge-Kutta

23. Proceed to:
Physics examples: Ideal harmonic oscillator – section

24. Physics Project – The pendulum, 13.7
I use a modified the C++ code from:
fout.close  fout.close()
Demo on folder:
\Lectures\05\CPP

25. ODEs in Matlab function dydt = odefun(t,y) a=0.001; b=1.0;
dydt =b*t*sin(t)+a*t*t;
Usage:
[t1, 100],0);
[t2, 100],0);
plot(t1,y1,’r’);
hold on
plot(t2,y2,’b’);
hold off
Demo folder: C:\Users\telzur\Documents\Weizmann\ScientificComputing\SC2011B\Lectures\05\Matlab

26. Output

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

33. In preparation Parallel execution
A new function (parallel_run) allows parallel computations and leverages multicore architectures and their capacities.
In future Scilab versions:

39. Xcos demo