12'010 Computational Methods of Scientific Programming

1
12.010 Computational Methods of Scientific
Programming

• Lecturers
• Thomas A Herring, Room 54-618, tah_at_mit.edu
• Chris Hill, Room 54-1511, cnh_at_gulf.mit.edu
• Web page http//www-gpsg.mit.edu/tah/12.010

2
Overview Today

• Review some aspects of Homework 4 (Mathematica).
• Look at formatting of question 1
• Look at FindRoot and NDSolve for Question 3.
• Final Project Examine Matlab N-body project to
  show
• methods Animation
• Solution using odeXX differential equation solver
  in Matlab
• End by looking at some graphics packages

3
Mathematica Homework

• Question 1 Formatting output was trickyThe
  notebook 12.010_Lec20_Mathematica.nb gives
  examples for some solutions
• Question 2 OK, features from 5.2 Mathematica
  used
• Question 3 NDSolve was the way to solve this
  problem along with FindRoot

4
Final Project

• Example case for N-body problem
• N-body planetary problem that use Runge-Kutta
  numerical integration with variable step size.
• Demonstrate basic use of this program (tar file
  on web site)
• Graphics usage in gui-form Basic methods can be
  used in your projects.

5
Graphics real-time output in Matlab

• In Matlab we can consider real time out put
  the results
• When a plot is made you can set an EraseMode
• Options are
• none leaves all points on the screen
• background paints old points with background
  color. Erases old points but also erases any
  other information such as grid lines and text
• xor Exclusive or. Erases just the previously
  plotted points, leaves the background intact

6
Generating animated sequence with Matlab

• Basic mode of use
• Plot first point keeping the graphics handle
• p plot3(x,y,z,., EraseMode,xor)
• Set the axis limits (Need to think of values to
  use here.
• axis(-100 100 -100 100 -100 100)
• hold on
• Now generate the sequence of points
• Loop over time steps, compute new x y z
• set(p,Xdata,x,Ydata,y,Zdata,z)
• Drawnow
• These ideas were demonstrated in Matlab M-file

7
ODE Solvers

• Use of ODE Solvers in Matlab (demonstrated in
  class)
• Vector y is 2-d position and velocity (14).
• y0 0.0 0.0 vx vz
• t,y,te,ye,ie ode23(_at_bacc,01tmax,y0,options
  )
• The bacc routine computes accelerations. dy/dt is
  returned so that dy1d(pos)/dty3
  dy2y4 and dy3 and dy4 are new
  accelerations
• function dy bacc(t, y)
• acc Computes accelerations
• Options sets ability to detect event such as
  hitting ground
• options odeset('AbsTol',terr 1 1
  1,'Events','hit')
• function value,isterminal,direction hit(t,y)
• Value returns the height.
• Look through Matlab help and use demo program

8
Graphics Programs

• Basic data plotting programs
• There a number of programs which fall into this
  basic category
• KaleidaGraph commercial program runs on PC and
  Mac (150) (web http//www.synergy.com)
• Tecplot Available on Athena (Unix and PC)
  (1300) (web http//www.amtec.com/)
• Grace -- Similar to KaleidaGraph available free
  for Unix systems. The old version is called
  xmgr. (http//plasma-gate.weizmann.ac.il/Grace/
• Demo of basic features of these types of programs

9
Summary

• Looked at Mathematica homework solution
• Example of 3-D Numerical integration Project
• Graphics packages
• Matlab homework due Thursday Dec 1.