12.010 Computational Methods of Scientific Programming

Lecturers

Thomas A Herring, Room 54-618, tah@mit.edu

Chris Hill, Room 54-1511, cnh@gulf.mit.edu

Web page http://www-gpsg.mit.edu/~tah/12.010

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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

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Mathematica Homework

• Question 1: Formatting output was tricky:The 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

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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.

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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

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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

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ODE Solvers

• Use of ODE Solvers in Matlab (demonstrated in class)• Vector y is 2-d position and velocity (1:4).y0 = [0.0; 0.0; vx; vz];[t,y,te,ye,ie] = ode23(@bacc,[0:1:tmax],y0,options);

– The bacc routine computes accelerations. dy/dt is returned so that dy[1]=d(pos)/dt=y[3]; dy[2]=y[4]; and dy[3] and dy[4] 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

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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

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Summary

• Looked at Mathematica homework solution• Example of 3-D Numerical integration Project• Graphics packages

• Matlab homework due Thursday Dec 1.