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PARTIAL DIFFERENTIAL EQUATIONS

There are three main types of partial differential equations.

  1. PARABOLIC: first order derivative in one variable, second order derivative in others, for example diffusion and the time dependent Schroedinger equation.

  2. ELLIPTIC: second order derivatives in all variables, but with the same signs, for example Poisson, time independent Schroedinger equation

  3. HYPERBOLIC: second order derivatives in all variables, but with different signs, for example the wave equations.

For this topic the best reference is Koonin, chapters 6 and 7. We will deal mainly with chapter 7, concerning parabolic differential equations. Other good references include numerical recipes, and ``Computational methods in Physics and Engineering'', by S. Wong.

Lets look at examples of parabolic P.D. E. s.

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A simple method to solve these will now be shown.

Example: diffusion with a uniform diffusion constant for 0 < x < 1, assuming Dirichelet Boundary conditions (specify phi on a large closed curve in the x,y plane) and specify the value of the field at the end-points of the interval, giving

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Make two approximations:

  1. spatial derivative approximated by finite differences on a uniform lattice of N+1 points with spacing h=1/N
  2. time derivatives approximated by a simple first order difference formula with a time step of delta t.

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at i=1 and N-1 the equation involves Dirichelet boundary conditions specifying the values of phi at 0 and N. This is an explicit difference scheme, given phi at one time can get at another time from

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with the action of operator H defined by

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In our example S=0 and phi is zero at x=0 and 1. Lets start with a gaussian centered at x=1/2.

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The last two are image gaussians to ensure the boundary conditions at x=1 and x=0.

A program to do this is given is chap7a.f which is in the week11.tar file. Run this for the following sets of values of time-steps and total time values: (0.00075 0.015), (0.00075 0.045), (0.001 0.02), (0.001 0.04), (0.001 0.06). These program writes its output to a file fort.11 which can be graphed using cplot.ex from lecture 2. (Download here if you already lost the file or switched computers. To use PGPLOT,
write pgplotcl

The analytic solution to this is

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At small timesteps the solution agrees with the analytic one. For which values is the solution stable? When does it start to become unstable? If the time step is very small the solution is complete on the scale of t near to 0.01. What happens for slightly larger timesteps? When do things start to go wrong and an unphysical stability develops. Details of the instability are given in the handout. See also the figure. The equation 7.4 of the handout is our equation 4 above, 7.5 is our equation 5.

Implicit schemes are one way around the instability. These are discussed in the handout. Results of the implicit scheme are shown in the last frame of the figure in the handout.

Another approach is the use of iterative methods, discussed in the handout.

We will finally explore the solution of the Time-Dependent Schroedinger Equation. The animated version of this routine can be downloaded as pspage.f here. this differs from the program distributed in previous years. The changes are moving the initiation of the window to a subroutine and adding the line call pgpap(6.0,1.0) before call pgenv(...).

This file also needs param.e7.


The original motivation for this project came from a page in the then current Physics 3H text by Tippler. See here for the full, full-circle, story of how this page began from movies in 1967 - probably at the same time as I was learning about Heisenberg back in High School in Australia. Local links here and here.
Videos in mpeg form based on this can be found at the physics simulation and animation website for supporting material for undergraduate physics teaching. This website is in development, including projects from this class. To date a section on elastic collisions (hebrew and english) and a section on Schroedinger sequations (english) have been completed. Please recommend this material if you are the metargel for Physics 1 (elastic collisions) and Physics 3 (Schroedinger equations.)

The animated gif file used in the elastic collision page was prepared by Yigal Ultchin using Animation Shop 2.

An easy way to make animated gifs in LINUX is to prepare a set of say *0001.png files (where * is say pics) and then write
convert pics*.png pics.gif
good if the files are relatively light.


Back to Lecture 11 page