The Art of Computational Science

© 2003- Piet Hut and Jun Makino

The Art of Computational Science series will provide a student with a hands-on guide to building a computational laboratory, and doing state-of-the-art research with it. The series will be self-contained: a high-school student should be able to start at page 1, and work her way through the series.

It may seem like a tall order, to use a series of books to lead a student through a process that normally takes five years, from entering college to being able to do research as a graduate student. What we offer here is a shortcut, not a replacement of a traditional curriculum. A motivated high school student or undergraduate could set aside a summer for self-study, or even better, pair up with another student, to start working through this book series. That way, they would quickly get a taste for conducting real independent research. Having acquired the taste and experience will be invaluable for them, no matter what speciality they would choose in the end.

The problem with the hundreds of introductory text books to science is that they mostly provide summaries, highly distilled collections of knowledge that can only be internalized through a process of hands-on experience that is generally left out. Even when detailed exercises are given, together with their solutions, this still leaves out the actual experience of anybody trying to solve a problem, and getting hopelessly astray. Research is (much) more than 90% about making mistakes, and (much) less than 10% about finding something new and original. Educational learning is no different. We all learn from solving excercises, not by the answer we get, but by learning what not to do wrong in the process.

Of course, there is a good reason that text books are presented in the style they are presented. Giving the reader a real-life impression of all (or at least much) that can go wrong in solving problems would expand the size of a book by a factor of ten, if not more. Therefore, it is certainly a good thing that most text books are as they are, providing reference material for a course, and providing the kernel of knowledge for gifted or already somewhat experienced students who are brave enough to try to learn the material on their own.

However, we think there is room for a different approach, one that has not been attempted earlier, as far as we know. We will try to follow a few individual students, getting occasional guidance from a teacher, in the actual process of learning through trial and error. This choice dictates the format as that of a dialogue, in which we can overhear what goes wrong, and how the students sooner or later find out how to correct their errors and misunderstandings.

Still, we are faced with the problem of cramming five years of student experience in only ten volumes. Our solution has been to take as simple a problem we could think of within science. In fact it is the oldest unsolved problem ever since the day Newton formulated mathematical physics, as the basis of modern science. It is the gravitional many-body problem, in which a collection of celestial bodies move under the influence of their mutual gravitational interactions.

An example of the two-body problem is the motion of the Moon around the Earth, or that of the Earth around the Sun. An example of the three-body problem is that of the Earth, Moon, and Sun now taken together, with the perturbations of the Sun taking into account in following the Moon on its way around the earth. The Sun and the major planets, from Mercury to Neptune, offers an example of a nine-body problem. A rich star cluster, such as M15 or 47 Tuc, provides an example of a million-body problem. And our whole galaxy, as well as our neighbor the Andromeda galaxy, can be modeled as a multi-billion-body gravitational problem.

Having selected a problem that can be explained relatively easily, we plan to introduce the notion of a physical force, the notion of a gravitational force, the idea of a differential equation, as well as the concept of numerical approximations to its solution. This will all appear in Volume 1. Hands-on experience in simulating the dance of two or three bodies will be the topic of Volume 2, while the study of a 32-body system, as a model for a star cluster, will occupy Volume 3. The current manuscript provides a first draft for these three volumes combined, but as yet without the most introductory steps; we hope to provide those soon(ish), in the next few years.

Volume 4 will introduce the notion of individual time steps, a crucial ingredient for simulating larger numbers of particles. We will also focus on a number of technical details in computer science, such as the introduction of self-describing flexible data formats, the use of classes in C++, XML of I/O, more complex command-line arguments, etc. This approach will show how to build robust modules that are flexible enough to function as tools in a growing toolbox that can used to build your own software environment tailered to the type of scientific simulations you would like to perform. We will use these tools to model 256-body systems.

Volume 5 will introduce the last major ingredient needed to model large systems, namely special treatments of tight binaries. With this in place, we can go to 2048-body systems (each increase of a factor eight in 3D doubles the number of particles in each spatial dimension). We will explore applications such as core collapse, and while doing so, we will find the need to add additional ingredients, such as a special treatments of triple star systems.

Volume 6 will introduce a diversion: it will focus on three-body scattering experiments. Just as high-energy particle accelerators probe the behavior of subatomic particles, we can probe the complex interactions between single stars and double stars in the virtual laboratory of the computer, by sending `beams' of single stars to (gravitationally) crash into `target plates' of double stars. These experiments are useful to predict and analyse the `microscopic' processes that happen within a star cluster scales small with respect to that of the cluster as a whole.

Volume 7 will continue the scattering treatment, but will handle the scattering of two double stars off each other, as well as the scattering of doubles and triples, and even more complex varieties, all of which occur within a large-scale simulation of a dense stellar system. As we will see, the automatization of such experiments, together with the automated reporting of the many different outcomes, will provide quite a challenge.

Volume 8 will pick up the main line of our series, the development of N-body codes with increasing sophistication (for historical reasons, the gravitional many-body problem is often called the N-body problem). This time we will start from scratch, using all the experience we have collected in volumes 1 through 7, by making a top-down design in terms of modules, while giving a complete specification of the interfaces between all these modules. This approach will allow us to mix and match different pieces written by different people, using different computer languages, and even adding different physical effects.

Volume 9 will continue this quest, by showing how such a code can be made to deal with the challenge of following close encounters, as well as tight double stars. Here we will learn to deal with various types of special treatments of neighboring particles, which is the main reasons that state-of-the-art N-body codes have grown so bulky. We will apply this code to some 16k-body systems.

Volume 10 will wrap us the quest, by introducing a way to handle even collisions between stars, through simple forms of hydrodynamics and stellar evolution. Depending on the speed of computers by the time we finish our series, we may be able to illustrate our code by taking two more steps of two in each dimension, leading us to the 128k-body system as well as the million-body system.

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