Reprint from 20th World Conference on Open Learning and Distance Education, Duesseldorf, Germany, April 2001

Teaching Numerical Methods for Partial Differential Equations over the Internet

André Jaun1,2, Johan Hedin1, Thomas Johnson1, Michael Christie3, Lars-Erik Jonsson4, Mikael Persson2

1 Alfvén Laboratory, Royal Institute of Technology, 100 44 Stockholm, Sweden
2 Electromagnetics, Chalmers Institute of Technology, 412 96 Göteborg, Sweden
3 Center for Educational Development, Chalmers Institute of Technology, 412 96 Göteborg, Sweden
4 Unit for Pedagogy and Didactics, Göteborg University, 412 96 Göteborg, Sweden

Email of corresponding author: jaun@fusion.kth.se

Abstract. A course has been organized in a virtual university environment involving students from the Royal- and the Chalmers Institute of Technology in Sweden and can now be taken at a distance with supervision on demand over the Internet (http://pde.fusion.kth.se). Short video conferences (synchronized with regular lectures) and video recordings (on demand for individuals) introduce the subject by following the teacher's line of thought; the material is then studied and assimilated in a second active learning phase, by performing numerical experiments in a Java powered web compendium. This enables every participant to work at his own pace, to develop intuition and assess his own understanding of finite difference, finite elements, Fourier, Monte-Carlo and Lagrangian methods. Assignments are carried out in a web browser and are automatically compiled into web pages where the students explain with their own words, equations and programs how to derive, implement and run numerical schemes. Our experience shows that the technology is rapidly acquired by the context and that discussion forums provide an efficient framework letting not only the teacher, but also the students answer questions and discuss relevant topics. Examples with editable parameters are given for the advection, diffusion, Black-Scholes, Burger, Korteweg-DeVries and Schrödinger equations: each can be used to illustrate fundamental concepts in a range of subjects dealing with the pricing of stock market options, the transport of heat or the scattering of a particle in quantum mechanics.

Introduction.

Numerical methods... this sounds like a perfect topic to teach using a computer! Yet too often, courses in computational methods focus on the mathematics of the teacher's favorite and miss both an overview and examples that are needed by the students to implement an optimal solution for a specific problem. Convinced that the Internet technology can be useful in this context, with Johan assisting and Thomas joining a class of sixteen mostly graduate students dispersed between Stockholm and Göteborg, we decided to embark on a new form of electronic teaching and carried out pedagogical experiments during the summer of 1999 and 2000.

The concept

combines short classroom lectures that are well suited for video conferences with an individualized active learning phase based on a Java powered web compendium. The students carry out home assignments in a web browser and submit them for correction to the teacher and assistants. News groups and discussion forums exploit the ability of quick learners to answer simple questions from peers and enable the teacher to focus on problems where his expertise is most useful and precious. Being active researchers in theoretical plasma physics, it is clear that we could only devote a limited amount of time to the entire project. How much can be achieved in a total of six months work, how large is the technology burden and how useful are the electronic tools in this context was largely unknown and was part of the experiments. Together with the material that we developed, it is the positive experience and the encouraging results we obtained that we would like to share here with you.
All begins with an outline sent to the school's mailing lists (students97@engineering.myschool.edu, phd-students@myschool.edu) announcing the course. A link to the web compendium and former student projects enables potential participants to judge if the content is aligned with their target curriculum. This way of proceeding reflects the current trend towards a free market for university courses and is well adapted to offering teaching services outside traditional school boundaries, such as sister universities and private companies. But is it reasonable to expect that students visit universities and even pay for courses if the compendium is readily accessible for free over the Internet? Our experience shows that those students who have the possibility to attend classes still do so for the stimulation and discussions they get directly from the teacher and peers. Those who cannot strongly value the flexibility of delivering exercises when and where they like and nevertheless to benefit from a personal supervision and corrections from a teacher.

In the classroom,

traditional ex caetedra lectures introduce the web page equivalents of the compendium [1] and are broadcast by video conference to remote participants. The lessons are short (30-40 minutes) to maintain the attention of the audience and are organized to stimulate discussions locally before sharing the conclusions from remote classrooms. The JBONE applet (Java Bed for ONE dimensional problems) is used to test every new scheme directly in the web browser; this adds an unprecedented animation and interactivity to the lecture and is extremely valuable when comparing the numerical properties of different time evolution schemes. Menus select the equation (advection diffusion, Burger's shock waves, KdV's solitons, Black-Scholes options, Schrödinger), the initial condition (box, Gaussian, cosine, soliton, put option, wavepacket) and editable text fields control the parameters (velocity, diffusion, dispersion, time step, etc.) directly in the web browser. A mouse click starts the simulation, making it extremely easy and convincing to illustrate for example the linear instability that occurs when the time step is too large in an explicit finite difference advection scheme, or to show how the more subtle aliasing in spectral methods non linearly affects a train of colliding solitons.

At home.

An advantage of using widespread, platform independent technology is that the students can reproduce and modify the demonstrations back in their office or directly from home. Repeating slowly the line of thought from the classroom, the text and figures take the reader through the series of analytical derivations resulting in a numerical scheme. Hyperlinks point to the relevant sections in the code and show practically how every algorithm has been implemented. Default parameters are preset to illustrate specific properties, but can be temporarily modified to verify if the topic has been properly understood.
Example: from Brownian motion to diffusion. Both are fundamental in science and engineering and are often hard to understand for undergraduates. An analytical derivation of the RMS displacement Ö< xi2 > in a particle's random walk (connecting the diffusion coefficient to the square of the mean free path divided by the collision time D = lmfp2/2tc might en-light a few, but is likely to loose a majority in the algebra. Using the Monte-Carlo evolution from the applet displayed in Figure 1, it is simple to demonstrate first how the random motion of a single particle can be described with the simple algorithm 
   for (int j=0; j<numberOfParticles; j++){
     particlePosition[j] += 
        random.nextGaussian() *
        Math.sqrt(2 * diffusCo * timeStep);
   }
Distributing 1000 particles alike to approximate an initial Gaussian distribution, the same algorithm immediately shows what diffusion really means! Take a different initial condition such as a cosine or a square box and it is equally rewarding to demonstrate how short wavelength modes decay more rapidly than long wavelengths.

FIGURE 1. Screen capture of the web browser displaying analytical formulae, the algorithm with a hyperlink into the source code and the JBONE applet -- here after execution of the Monte-Carlo integration with 1000 particles to illustrate the connection between Brownian motion and diffusion.

Having part of the students who study away from the campus, is it possible to use the technology to stimulate interactions with the teacher and peers? Yes, better than we thought!

The assignments.

Exercises in the first session are designed to familiarize the students with the web technology, using templates to show the building blocks are used in analogous situations to gradually assimilate LATEX and Java directly in the context. Teaching at an advanced level where copying is not an issue anymore, we decided to distribute a list of all the solutions web pages and let the students compare and discuss the results with each other. Our top pick of the best created a healthy competition, where everybody tried to become a member of a very exclusive list.
Part of the students chose to carry out an additional one week project, applying their favorite method to a topic of interest such as the Black-Scholes equation for a European call option, a tunable finite elements integration for the Schrödinger's equation, a mesh refinement procedure, iterative solvers, etc. Given the small amount of time allocated for such projects, the scope remained naturally limited; by cross-checking each other's reports on the web, the students however got an overview of a rather broad range of applications.
The material. A single LATEX source generates both the printed compendium and hyperlinked web pages. Running open software translators such as latex2html [2], tth [3] and scripts embedded in a makefile, the static web material can effectively be produced at no additional cost to what is anyway required to print a compendium and slides.
Writing the JBONE applet from scratch was quite an effort for the teachers, but the object oriented language and the encapsulated structure of the code enables students with no programming experience before the course to gradually modify existing schemes and finally add new ones. A substantial amount of documentation (programming tree, keyword index) is created automatically using the javadoc utility which is part of the standard java development kit. An automatic download service has been set up for teachers and individuals who would like to use, modify and tailor our material for their specific needs.

Discussion forums

News groups and discussion forums prove to be an ideal tool allowing a sufficient number of participants to interact in a geographically distributed environment. Not only do students help and discuss with each other at virtually any time of the day and night, but the advice is usually helpful and competent. Some supervision is of course required, but instead of answering a dozen times the same question (often for organizational matters), the teacher can intervene only once for an announcement and spend the rest of the time clarifying the discussions that remain very informal, helping the students to understand the subject in their own language.

Evaluation.

Considerable steps have been taken since the course was first taught in a conventional manner in 1997; they are the fruit of four years experience with students feed-back. Interactive lecture notes are relatively easy to introduce and can first be distributed as is only to supplement regular lecture notes. From a pedagogical point of view, the largest benefit however comes from introducing a problem based learning environment. This is much more delicate to implement with remote participants, when different computer platforms, software versions and operating systems quickly become a technology nightmare! Our initial setup for example required that every student opened an X-terminal to connect to his UNIX account, compile Java code and publish the solutions manually on the web. In retrospect, it is largely thanks to Johan's broad knowledge and competent advice that most of the encryption and firewall problems could be solved smoothly during the first exercise session. The technology burden has been strongly reduced since by using a web browser for all the tasks, browser which is in fact tested before the course starts if the students register electronically.
One third of the participants in 1999 and 2000 filled-in an anonymous evaluation form. The overall impression they got from the course was good to excellent, one student judging that some of the web technology was kind of a burden and a large majority being very favorable to learning it by context. Suggestions and comments show that templates provide the most efficient help for the electronic submission of the home assignments. The questions asked in the discussion forums concerned mainly computational aspects (i.e. the substance of the course). One participant proposed to create a discussion group reserved for course alumnies to maintain valuable contacts even after graduation. In its full electronic form, the course clearly requires a well maintained web server, which is generally administered by an assistant in a university. Because of the ask once, answer to all nature of the discussion forums, assistants can however be employed very efficiently, so that the overall teaching load is finally similar to a conventional setup.
Some flexibility is required from the lecturer and the students to exploit the possibilities and work around the weaknesses of a course taught at a distance over the Internet. Our experience however shows that the pedagogical content is by no means reduced if the technology can be used to create a problem based learning environement, where discussion forums enable students to interact and understand new topics with their own words. The enthusiasm from all the participants including the teaching assistants is a very gratifying experience and should be an encouragement to try similar experiments in other fields.

Bibliography.

  1. A. Jaun, J. Hedin, T. Johnson, Numerical Methods for Partial Differential Equations, TRITA-ALF-1999-05 (1999)
  2. N. Drakos, Text to Hypertext conversion with LaTeX2HTML, Baskerville 3 (1993) 12
  3. I. Hutchinson, TTH: a TeX to HTML translator