Homepage of John C. Butcher


Welcome to the homepage of John Butcher of Auckland, New Zealand.

I am a mathematician specializing in numerical analysis.

Short biography

  • Born in Auckland, New Zealand
  • Educated at various primary schools in Auckland and in Dargaville
    primary school
  • Secondary education at Dargaville DHS, Taumarunui DHS and Hamilton HS
  • Tertiary education at Auckland University College and University of
    Sydney. BSc and MSc (Auckland); PhD and DSc (Sydney)
  • academic and research positions at Sydney and Canterbury
    Universities and Stanford Linear Accelerator Center
  • Professor of Mathematics, University of Auckland 1966 - 1998
  • Emeritus Professor 1999 - present

Research Interests

Ordinary differential equations

Numerical methods for the solution of ordinary differential equations
(section 65L in Mathematical Reviews).
\[ \frac{d y}{d x} = f(y) \]
Within this broad area I work on a number of specialities some of which
are listed below.

Runge-Kutta methods

\[ \begin{array}{c|cccc} c_1 & a_{11} & a_{12} & \cdots & a_{1s}\\
c_2 & a_{21} & a_{22} & \cdots & a_{2s}\\
\vdots & \vdots &\vdots & &\vdots \\
c_s & a_{s1} & a_{s2} & \cdots & a_{ss}\\ \hline
& b_1 & b_2 & \cdots & b_s \end{array} \]

General linear methods

\[ \begin{bmatrix} A & U\\ B & V \end{bmatrix} \]

G-symplectic general linear methods

\[ \begin{bmatrix}
DA + A^{\sf T} D - B ^{\sf T} G B & DU - B^{\sf T} G V\\
U^{\sf T}D - V^{\sf T} G B & G - V^{\sf T} G V \end{bmatrix}=0. \]

Honours and awards

  • Fellow Royal Society of New Zealand, 1980
  • New Zealand Mathematical Society, Research Award
  • Hector Medal, RSNZ, 1996
  • Life Member, NZMS, 1998
  • Fellow, NZMS, 1999
  • ICCMSE Prize for Computational Mathematics, 2003
  • Honorary Fellowship: European Society of Computational
    and Applied Mathematics, 2008
  • Fellow, Society for Industrial and Applied Mathematics, 2010
  • Jones Medal, Royal Society of New Zealand, 2010
  • van Wijngaarden Award, 2011
  • Officer of the New Zealand Order of Merit, 2013

Travel and conferences 2015


Beijing, 10-14 August 2015

Contributed paper: "On fifth order Runge-Kutta methods"


Kuala Lumpur, 28 August 2015

Invited paper: "Counting trees and rooted trees with applications"

Invited workshop: "B-series for the analysis of numerical methods"

Universiti Putra Malaysia

Putra Jaya, 17-28 August 2015

Three lectures on Runge-Kutta methods (17 - 19 August)

Seminar (28 August): "Solving evolutionary problems stably and conservatively"

Runge-Kutta workshop

Kassel, 31 August 31 - 2 September 2015

Invited presentations: “Survey of Runge-Kutta Methods“, "Runge-Kutta Methods for Non-Stiff Problems", "Runge-Kutta Methods for Stiff Problems", "Runge-Kutta Methods for Hamiltonian Problems"

Humboldt Universiät zu Berlin

Berlin, 4 September 2015:

Seminar "B-series and the order of general linear methods"


Halle, 7-11 September 2015

Contributed paper within Minisymposium in honour of R. März: “Constructing high order G-symplectic methods“


Potsdam, 14-18 September 2015

Contributed paper: “The cohesiveness of G-symplectic methods“

Universiät Kassel

Kassel 21 - 23 September 2015:

Seminar "B-series for the analysis of numerical methods"

Book: "Numerical methods for ordinary differential equations"

Third edition

I am now working on the third edition. Any comments on the previous
editions that will help me make improvements would be very much appreciated.

My University of Auckland web-site

At the present time, information is being shared, and in some cases duplicated between this site and my site at the University of Auckland.