John Butcher talks to friends about numerical ordinary differential equations

February 2008 has turned out to be a great month for me.  Not only did the new edition of my book go into production, but I have had an extraordinarily large number of people with me to talk about numerical methods for differential equations.  Let me say who these friends are,  approximately in the order in which they first made contact with this subject.

First there are Allison Heard and Robert Chan, each a PhD graduate in numerical analysis at the University of Auckland, and each a colleague in the Department of Mathematics.

Then there are Tina Chan and her husband David Chen.  They each did PhDs in Auckland and each of them is now a professor in Taiwan.  It is fortunate for me that they are visiting Auckland this month.

Shixiao Wang is a colleague in this department.  Although his main interests are in fluid mechanics, he has become an active member of the weekly numerical analysis workshop.

Helmut Podhaisky and Daniel Weiß are PhD graduates at the Universities of Halle and Cologne respectively.  Helmut visits Auckland quite often and is here this month.  Daniel, who works in differential-algebraic equations, is here for all of 2008.

Angela Tsai, Maria Goodier and Yousaf Habib are current PhD students in this department.

Jane Lee and Dawoomi Kim have studied specialist courses in numerical analysis in Auckland.  Jane hopes to travel overseas to study this subject at a higher level.  Dawoomi has other interests as well and will, at the end of February, leave us for a position outside the University.
I asked each of my friends why they have chosen this subject to study as far as they have and why, in many cases, it has become a lifelong interest. After reporting each of their answers, I will give my own answer to the same question.

John and friends

Allison Heard Allison:

During my undergraduate degree I found the analytical solution of differential equations interesting.  As part of a graduate paper on numerical analysis, taught by John Butcher, we were required to do a project on a topic of our choice. This seemed a perfect opportunity to do some work on differential equations. It wasn’t until I was searching the literature that I discovered that John was prominent in the area of numerical solutions for ordinary differential equations.  I enjoyed that project and so became interested in that area of numerical analysis – and interest that continued while I was a junior lecturer.
John did suggest that I might do a Ph D in numerical analysis but instead I went to Britain on a working holiday, which included office work and teaching in secondary schools. During that time, I became attracted to studying for a Ph D, and spent some time reading Henrici’s book, Discrete Variable Methods in Ordinary Differential Equations. On my return to Auckland, I enrolled in a Ph D with John.
Numerical methods for ODEs brings together pure mathematics, in the theoretical basis for methods and their behaviour, and practical applications, since ODEs are widely used in modelling. I find this very satisfying. Since that time, apart from raising a family, I’ve been involved in John’s group at Auckland University.
John has been very supportive of me. Working with him means that you have a steady stream of good ideas readily available, as well as someone to help when it is needed.

I was lucky to be able to choose what I wanted to study and with whom. It did not take me long to decide on numerical analysis for my subject and John Butcher for my supervisor.
I was also fortunate that John was so welcoming that there were no hurdles for me to jump before being accepted. I chose to study with John because he was such an awe-inspiring mathematician and I am so lucky to have the privilege of seeing the development of not only his mathematics but his endearing humanity at first hand.
As for my choosing to work on numerical methods for ordinary differential equations, this was undoubtedly the right choice for me and one I have never regretted.
Robert Chan
Tina Chan Tina:

It was an inspiration to be supervised by a pioneer in this field. I worked on applications of the Algebraic Theory of Runge-Kutta Methods and I found the recursive formulation of the group operation to be particularly interesting. The work was difficult but it was really worth the effort.  Recently Butcher's algebraic structure has been identified as a Hopf Algebra and it has applications in Physics and elsewhere. But even in its original role in numerical analysis, and through its alternative formulation as B-series, it still has new applications and generalizations and is still a powerful area of study.
David Chen David:

From studying this subject, I learn how to simplify some tedious computing tasks. I also learn programming and computing skills from doing experiments with some computational software. It is very useful to have implementation in many fields. In particular, I take the advantage from staying on the shoulder of a giant.

In simulating complex flows such as vortex breakdown and turbulence, the efficiency of the numerical methods at the heart of the simulation is the key to success. This is only possible by developing high order numerical methods. The modern theory of Runge-Kutta methods, based on the use of trees and originally developed by John Butcher, is at the very heart of this subject.  In my view, this is a must-learn theory for anyone engaged in the study of complex flows through computer simulation. My personal experience confirms this: my research has greatly benefited from learning this beautiful subject.
Shixiao Wang
Helmut Podhaisky Helmut:

Prof. Rüdiger Weiner introduced me to the subject when I was a undergraduate student at the University of Halle in 1995.  Interestingly, this was the year in which the textbook "Numerik gewöhnlicher Differentialgleichungen", by Strehmel and Weiner, was published. At that time, I didn't know much about differential equations, but I felt that it would be very desirable to join what was a very active research group.
I soon began to like the subject, because of the combination of computation, analysis and  even discrete mathematics which were intrinsic to its study. A very beautiful result, and an impressive example of this diversity, is the connection between rooted trees and the accuracy of numerical methods.

I am fascinated by the theory of differential- and differential-algebraic equations due to their various applications in physics, biology, medicine and engineering.  But in most cases modelling alone is not sufficient, and numerical methods are required.  Developing and studying numerical methods to solve these equations is a very exciting and dynamic area of applied research. Many different questions and problems from different parts of mathematics, and even from computer science, have to be solved.
I am glad to be in Auckland with John Butcher, a person who has contributed a great deal to this research.
Daniel Weiss
Angela Tsai Angela:

For me, the door of numerical methods for ODEs was opened by Robert Chan and John Butcher at the time when I took papers and did a project as part of my Bachelor of Technology degree in Industrial Mathematics.  It fascinates me that there are so many different areas of interest in numerical analysis.  I enjoy both learning the theoretical side of numerical methods and testing them out on computer.  It's really great meeting and exchanging ideas with people from different parts of the world, who have come to Auckland as visitors or to participate in our conferences and workshops.  And of course, we have such a nice group of people, the members in our group are my 'family away from family'.
Maria Goodier Maria:

Differential equations model many useful life processes, and creating efficient ways of solving them represents scientific progress.  I also enjoy writing programs and making them work.  It is great being a part of the research group that we have here in Auckland.

I was interested in learning about the applications of Mathematics in the real world. In the final year of my under graduate study, I took a course on Fluid Mechanics which was wonderfuly taught by Dr. Rafiq. I had a chance to apply the numerical techniques to the equations in Fluid Mechanics. That motivated me to explore more about the field of Numerical Analysis. Later I studied for an advanced Master's course where I took courses on Slow Viscous Flows and Computational Fluid Mechanics. I was fortunate to have attended the course on Advance Numerical Analysis as well. This was a fun and interesting course taught by Prof. Arieh Iserles. I enjoyed it so much that I decided to take on my Master thesis with him. I have since been trying, successfully or unsuccesfully to learn more about the Numerical Analysis. I am recently working under the guidance of Prof. John Butcher and I have been able to explore new dimensions in the field of Numerical ODEs.
Yousaf Habib

I began my undergraduate studies in physics and soon I realised how important mathematics is in understanding physics. So, I took more mathematics papers and I soon became interested in numerical analysis. It was very exciting to learn different techniques for solving scientific problems and it was even more fun to test these on the computer. I had no doubt that this field has an important role in many science areas, because, these days, computer models and simulations are used in all fields of scientific research.
During the last year of undergraduate study, I was very fortunate to meet Prof John Butcher as one of lecturers in a numerical methods course. At first, I was simply surprised by the rooted trees and how they represent Runge-Kutta methods but, as time went by, I could see that how the theory of rooted trees covers wider and deeper analysis of numerical methods. So, I decided to carry on my study to master level to learn more about numerical methods for ODEs. Now, I look forward to researching more on this exciting area which is fundamental in a wide range of applications.
Jane Lee
Dawoomi Kim Dawoomi:

Numerical Analysis was a key subject for studying applied mathematics.  I took a specialist course in Numerical Analysis in 2006.  I did not have much knowledge about this subject when I started this course, but I enjoyed working on the higher order of numerical methods for finding accurate solutions as I began to understand the subject.

Thank you all for your answers.

Perhaps I might say how I got into this subject, which I have worked on, in one way or another, as long as almost anyone else in the world.

I went to a conference in Australia in 1957 when I was a Physics PhD student. Many famous numerical analysts spoke and I started to understand the beauty and elegance of the subject, as well as its growing importance in the computer  age which was dawning at that time. Two of the speakers, S. Gill and H. Merson, had made important contributions to Runge-Kutta methods and this topic and their work especially intrigued me.  At that stage of my life I was too shy to talk to either of them but I followed up on some of their ideas and eventually got my 1963 paper published. I realised that Mathematics, and not Physics, was what I should be doing and this is how I have spent most of my time since then. My own knowledge of modern Mathematics was severely limited when I started, but numerical methods for differential equations has been a great place for me to learn more about the varied and beautiful tools used in this subject.
John Butcher