Using Computers to Build Insight
Extracts from the proposal
Laboratory for Computational Physics:
Using Computers to Build Insight
submitted to the NSF Instrumentation
and Laboratory Improvement program
We propose to offer a new course entitled "Introduction to Computational
Physics." This course takes advantage of the many cases in which the
physical question and its algorithmic solution are intimately linked, so the
physics and the algorithm illuminate each other. Because it exploits this
link to the fullest, the course has only one prerequisite physics course, and
thus will be taken by students as early as their freshman year. The course
will 1) excite and attract students at an early stage in their college
careers, 2) allow appropriate use of computation throughout the physics
curriculum, and 3) develop the students' physical insight.
A. Current Situation
Oberlin College offers the standard sort of graduate school preparation
curriculum that can be found at any top-notch college or university. We do
not want to disparage this curriculum: it has become standard precisely
because it has proven itself so successful. However the standard does have
widely acknowledged flaws, such as:
- Students must labor through four hard years before they begin to see
current research applications in their formal course work. Many students
drop out along the way.
- The standard curriculum solidified before the advent of the
this most practical of physics tools is largely ignored. At Oberlin, for
example, physics majors are urged to learn a computer language, yet most of
those who do so apply this knowledge only to one or two relaxation problems
that might or might not be assigned in the electrodynamics course.
- The curriculum tends to produce graduates who manipulate
than understand physics. For example, any physics senior will be able to
express a periodic function as a Fourier series with frequencies going from
+\infinity to -\infinity, but few will be able to explain the meaning of
the negative frequencies.
This proposal is an attempt in the direction of rectifying the faults
while retaining the manifold and obvious benefits of the standard curriculum.
B. Course Philosophy
The typical computational physics course is offered to juniors or
seniors as a "value added" course: The students are already familiar with
the physical problems and with analytic methods of solving them, and they are
taught a new set of methods, the algorithmic methods, to glue onto those
already known. Because this course comes so late in the curriculum, it gives
the impression that numerical methods are second-rate and are to be
when analytic methods fail. This approach ignores the fact that, in many
cases, the algorithmic solution to a problem is directly related to the
fundamental concept of the problem, and thus a knowledge of the algorithmic
solution actually helps the student understand the fundamental concept. As
an example, consider the initial value problem dx/dt = x, x(0) = 1. The
analytic (symbol manipulating) method of solving this problem relies on facts
such as \integral dx/x = ln(x), which are conceptually very far removed from the
fundamental idea that the derivative is a slope. In contrast the algorithmic
method (say Runge-Kutta) is a direct embodiment of that fundamental idea. By
understanding the Runge-Kutta method and coding it into a program, the
student not only builds a practical tool for solving differential equations,
but also builds up his own understanding of the idea of a derivative. This
is a situation where teaching the algorithm at an early stage will aid the
student's conceptual development and will build his physical insight.
[Note: No one will deny the many advantages of analytic over numerical
solutions (it is, for example, impossible to choose the wrong step size in an
analytic solution), nor will anyone deny that sometimes the algorithmic
technique is not conceptually enlightening (e.g. the QR algorithm for the
eigenvalue problem). The approach advocated here will not cure all ills.
Nothing will. Instead, we expect the two sorts of instruction to complement
and reinforce one another.]
Our proposed computational physics course takes advantage of such
mutually advantageous situations to teach algorithms as aids for
understanding physics rather than as (somewhat distasteful) mechanisms for
getting answers. In order to develop the connection between formal
approaches and informal, physical (and largely pictorial) approaches, the
course will use computer graphics in an essential way. The course will have
only one prerequisite, namely the introductory "Mechanics and Relativity"
course (which in turn has a prerequisite of one semester of calculus). Thus
students will be ready for the course as early as the second semester of the
freshman year (if they arrive at college with calculus advanced placement, as
many do). In this position, we will use the course to inject excitement,
research problems, and physical insight early on in a physics student's
career. Also in this position, it will to useful to students in allied
sciences such as chemistry, computer science, and geology. We will select
examples to make the course attractive to such students.
C. Course Outline
The course consists of four equal sections: 1) groundwork, 2) ordinary
differential equations and chaos, 3) spectral analysis, and 4) harmonic
functions. The groundwork section presents instruction in the FORTRAN
language through straightforward applications to zero-finding in polynomials.
FORTRAN was selected over Pascal or C because i) it has a relatively
straightforward syntax, ii) it posseses the necessary language features to
support numerical work, and iii) it is the de facto standard for scientific
work. We will be able to condense the actual language instruction into three
weeks because many features, such as character variables, pointers, and
recursion, will not be taught.
The ordinary differential equation section will build directly
student's background in the "Mechanics and Relativity" course. The numerical
technique taught will be Runge-Kutta and examples will be drawn from the
field of chaos. This field will contribute excitement, of course, but is
also valuable in building a student's physical insight. Chaos relies heavily
on visualization techniques, such as the Poincare section, whose entire
purpose is to generate a good qualitative picture of the motion. These
techniques are also valuable for building qualitative understanding in
situations where analytic solutions are available. They are not commonly
used as such simply because the analytic solution is used as a crutch to
avoid a deep qualitative understanding of the situation.
Every professional physicist has a qualitative understanding
analysis. Students usually do not, because Fourier techniques are introduced
accompanied by such a cacophony of difficult integrals and convergence
theorems that the student misses the fundamental qualitative point. These
topics are essential for a second encounter with the subject, but for our
introduction we will stress examples and qualitative pictures. This section
will give the course a much-needed experimental component.
The last section would normally be called "solutions of Laplace's
equation," but our students will not even know what a partial derivative is!
We will instead speak of harmonic functions, defined at those
the mean value relation. It is no more mysterious to say that the
temperature of a plate must be a harmonic function than it is to say that the
temperature obeys Laplace's equation. The mean value relation leads directly
to relaxation techniques and then to random walk techniques. Multigrid
methods (which apply relaxation to grids of varying size) will be introduced
both to speed the computations and to reinforce the qualitative understanding
of wavelength components developed in the spectral analysis section.