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MA/CS 375

MA/CS 375. Fall 2002 Lecture 13. Office Hours. My office hours Rm 435, Humanities, Tuesdays from 1:30pm to 3:00pm Rm 435, Humanities, Thursdays from 1:30pm to 3:00pm Tom Hunt is the TA for this class. His lab hours are now as follow SCSI 1004, Tuesdays from 3:30 until 4:45

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MA/CS 375

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  1. MA/CS 375 Fall 2002 Lecture 13 MA/CS 375 Fall 2002

  2. Office Hours • My office hours • Rm 435, Humanities, Tuesdays from 1:30pm to 3:00pm • Rm 435, Humanities, Thursdays from 1:30pm to 3:00pm • Tom Hunt is the TA for this class. His lab hours are now as follow • SCSI 1004, Tuesdays from 3:30 until 4:45 • SCSI 1004, Wednesdays from 12:00 until 12:50 • Hum 346 on Wednesdays from 2:30-3:30 MA/CS 375 Fall 2002

  3. Review Ordinary Differential Equation • Example: • we should know from intro calculus that: • then obviously: MA/CS 375 Fall 2002

  4. Review Forward Euler Numerical Scheme • Numerical scheme: • Discrete scheme: where: MA/CS 375 Fall 2002

  5. Review Direct Proof of Convergence • Fix T and let delta t drop to zero • In this case n needs to increase to infinity MA/CS 375 Fall 2002

  6. Review Stable Approximations • 0<dt<1 dt dt=0.125 dt=0.5 dt=0.25 MA/CS 375 Fall 2002

  7. Review Application: Newtonian Motion MA/CS 375 Fall 2002

  8. m2 m1 Each particle has a scalar mass quantity Review Two Gravitating Particle Masses MA/CS 375 Fall 2002

  9. Review Particle Positions Each particle has a vector position x2 x1 (0,0) MA/CS 375 Fall 2002

  10. Review Particle Velocities Each particle has a vector velocity v1 v2 MA/CS 375 Fall 2002

  11. Review Particle Accelerations Each particle has a vector acceleration a1 a2 MA/CS 375 Fall 2002

  12. Review N-Body Newtonian Gravitation • For particle n out of N The force on each particle is a sum of the gravitational force between each other particle MA/CS 375 Fall 2002

  13. Review N-Body Newtonian Gravitation Simulation • Goal: to find out where all the objects are after a time T • We need to specify the initial velocity and positions of the objects. • Next we need a numerical scheme to advance the equations in time. • Can use forward Euler…. as a first approach. MA/CS 375 Fall 2002

  14. Review Numerical Scheme For m=1 to FinalTime/dt For n=1 to number of objects End For n=1 to number of objects End End MA/CS 375 Fall 2002

  15. Review planets1.m Matlab script • I have written a planets1.m script. • The quantities in the file are in units of • kg (kilograms -- mass) • m (meters – length) • s (seconds – time) • It evolves the planet positions in time according to Newton’s law of gravitation. • It uses Euler-Forward to discretize the motion. • All planets are lined up at y=0 at t=0 • All planets are set to travel in the y-direction at t=0 MA/CS 375 Fall 2002

  16. Mercury has nearly completed its orbit. Data shows 88 days. Run for 3 more days and the simulation agrees!!!. Earth Venus Review Sun Mercury MA/CS 375 Fall 2002

  17. Today Team Exercise • Get the planets1.m file from the web site • This scripts includes: • the mass of all planets and the sun • their mean distance from the sun • the mean velocity of the planets. • Run the script, see how the planets run! • Add a comet to the system (increase Nsphere etc.) • Start the comet out near Jupiter with an initial velocity heading in system. • Add a moon near the earth. • Extra credit if you can make the comet loop the sun and hit a planet  MA/CS 375 Fall 2002

  18. This Lecture • More accurate schemes • More complicated ODEs • Variable time step and embedded methods used to make sure errors are within a tolerance. MA/CS 375 Fall 2002

  19. Adams-Bashforth Schemes • In the forward Euler scheme we only used the value of the right hand side at the previous time step. • i.e. we only used a linear approximation to the time derivative MA/CS 375 Fall 2002

  20. AB Schemes • Idea: we set • where If interpolates fn,fn-1,fn-2,..,fn+1-Nstages • i.e.: MA/CS 375 Fall 2002

  21. f0 f1 f2 f3 We interpolate the function through thefirst 4 points. Then we integrate under the curve between t=3 and t=4. MA/CS 375 Fall 2002

  22. AB Schemes Essentially we use interpolation and a Newton-Cotes quadrature formula to formulate:

  23. Runge-Kutta Schemes • See van Loan for derivation of RK2 and RK4. • I prefer the following (simple) scheme due to Jameson, Schmidt and Turkel (1981): MA/CS 375 Fall 2002

  24. Runge-Kutta Schemes • Beware, it only works when f is a function of y and not t • here s is the order of the scheme. MA/CS 375 Fall 2002

  25. Error Estimate • Matlab has a number of time integrators built in. • ode23 • ode45 • and others.. MA/CS 375 Fall 2002

  26. ode23 • For n=1:#timesteps • ode23 uses two estimates for yn+1 . • A 2nd order RK scheme and a 3rd order RK scheme are used to build two guesses for yn+1. • If the difference between these two estimates are within a tolerance ode23 progresses on to calculating yn+2 • If the difference is greater than the specified tolerance, ode23 reduces the dt and tries again. It repeats until the difference is lower than the tolerance. End MA/CS 375 Fall 2002

  27. Planets Example Using ode23 • Idea: replace our home grown Euler Forward scheme with Matlab’s ode23 scheme in the planets1.m script. MA/CS 375 Fall 2002

  28. Parameters forthe planets asbefore MA/CS 375 Fall 2002

  29. Initial velocities Gather X,Y,VX,VY into one vector MA/CS 375 Fall 2002

  30. Call to Matlab ode23 function [T,CoordVel] = ode23(‘forcing’, TimeLimits, CoordVel(:)); function whichcalculates f(y) Vector holds X,Y,VX,VY [0 FinalTime] Works in Matlab v6.1.0 … at least MA/CS 375 Fall 2002

  31. Call to ode23 Find out the time at each time step Extract final coordinates and velocities Plot planet orbits MA/CS 375 Fall 2002

  32. The routine whichcalculates the forcing function. MA/CS 375 Fall 2002

  33. Team Exercise • Grab planets2.m and forcing.m from the http://useme.org/MA375.html • Run the script • Use the Tsteps vector to find out the time step for each integration stage. Plot a graph showing the time step (dt) at each time step. • Use help to find out how to change the tolerance used by ode23 (hint you will need to use odeset) • Rerun the simulation with a tolerance of 0.1 MA/CS 375 Fall 2002

  34. Application: One-Dimensional Electrostatic Motion MA/CS 375 Fall 2002

  35. Charge Repulsion • Now we will consider the case of charged particles with the same sign charge • Instead of attracting each other, the charges repel each other. MA/CS 375 Fall 2002

  36. Particle Accelerations Each particle has a vector accelerationdirectly away from the other particle a2 a1 MA/CS 375 Fall 2002

  37. Team Project Q1) Modify the planets2.m and forcing.m to simulate the following: • There are N electrically charged particles confined to move in the x-direction only • Distribute the charges initially at equispaced points in [-1,1] • The equations of motion of the charges are: MA/CS 375 Fall 2002

  38. Team ProjectReport Required on 9/25/02 Q1cont) Include the following in one project write up per team: • A scatter plot, with x as the horizontal axis and t as the vertical axis, showing the paths of all the charged particles using ode23 • A graph showing the size of each time step used by ode23 • Replace ode23 with ode45 and rerun • Plot the ode23 and ode45 (t,x) paths on the same graph. • Plot the ode23 and ode45 time step sizes on the same graph • Names of team members MA/CS 375 Fall 2002

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