John Butcher is recognised as the founder of the modern theory of Runge-Kutta methods and discover of the Butcher group. His contributions to numerical analysis include the formulation of implicit Runge-Kutta methods, particularly methods based on Gauss-Legendre, Radau and Lobatto quadrature, general linear methods, order barriers for explicit Runge-Kutta methods, effective order, non-linear stability and order arrows. His name is attached to the Butcher tableau and the Butcher product, as well as to the Butcher group and Butcher series.

John is a fellow of the New Zealand Mathematical Society, the Royal Society of New Zealand and the Society for Industrial and Applied Mathematics. He is an Officer of the New Zealand Order of Merit and his awards include the Jones Medal of the Royal Society of New Zealand and the Van Wijngaarden Award of the Centrum Wiskunde & Informatica, Amsterdam.

1 Differential equations, numerical methods and algebraic analysis.- 2 Trees and forests.- 3 B-series and algebraic analysis.- 4 Algebraic analysis and integration methods.- 5 B-series and Runge-Kutta methods.- 6 B-series and multivalue methods.- 7 B-series and geometric integration.- Answers to the exercises.- References.- Index.