John Charles Butcher ONZM (born 31 March 1933) is a New Zealand mathematician who specialises in numerical methods for the solution of ordinary differential equations. [1]
Butcher works on multistage methods for initial value problems, such as Runge-Kutta and general linear methods. The Butcher group and the Butcher tableau are named after him. More recently, he is investigating a new type of method with stability identical to that of a Runge-Kutta method.
| Positions held by John C. Butcher |
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Butcher studied mathematics at Auckland University College - BSc and MSc - and the University of Sydney - PhD (1961) and DSc. Positions held are as aside.
He was awarded the Jones Medal from the Royal Society of New Zealand in 2010, for his "exceptional lifetime work on numerical methods for the solution of differential equations and leadership in the development of New Zealand mathematical sciences." [1] In 2011, he received the Van Wijngaarden Award. [3]
In the 2013 Queen's Birthday Honours, Butcher was appointed an Officer of the New Zealand Order of Merit, for services to mathematics. [4]
In numerical analysis, the Runge–Kutta methods are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta.
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals.
In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution.
In mathematics and computational science, the Euler method is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis.
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions and a number of points in the domain, and to select that solution which satisfies the given equation at the collocation points.
In numerical analysis and scientific computing, the backward Euler method is one of the most basic numerical methods for the solution of ordinary differential equations. It is similar to the (standard) Euler method, but differs in that it is an implicit method. The backward Euler method has error of order one in time.
In mathematics of stochastic systems, the Runge–Kutta method is a technique for the approximate numerical solution of a stochastic differential equation. It is a generalisation of the Runge–Kutta method for ordinary differential equations to stochastic differential equations (SDEs). Importantly, the method does not involve knowing derivatives of the coefficient functions in the SDEs.
In mathematics, the Runge–Kutta–Fehlberg method is an algorithm in numerical analysis for the numerical solution of ordinary differential equations. It was developed by the German mathematician Erwin Fehlberg and is based on the large class of Runge–Kutta methods.
In numerical analysis, the Cash–Karp method is a method for solving ordinary differential equations (ODEs). It was proposed by Professor Jeff R. Cash from Imperial College London and Alan H. Karp from IBM Scientific Center. The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Dormand–Prince (RKDP).
In numerical analysis, the Dormand–Prince (RKDP) method or DOPRI method, is an embedded method for solving ordinary differential equations (ODE). The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth-order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Cash–Karp (RKCK).
In mathematics and computational science, Heun's method may refer to the improved or modified Euler's method, or a similar two-stage Runge–Kutta method. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. Both variants can be seen as extensions of the Euler method into two-stage second-order Runge–Kutta methods.
In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by Hairer & Wanner (1974), is an infinite-dimensional Lie group first introduced in numerical analysis to study solutions of non-linear ordinary differential equations by the Runge–Kutta method. It arose from an algebraic formalism involving rooted trees that provides formal power series solutions of the differential equation modeling the flow of a vector field. It was Cayley (1857), prompted by the work of Sylvester on change of variables in differential calculus, who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics.
Within mathematics regarding differential equations, L-stability is a special case of A-stability, a property of Runge–Kutta methods for solving ordinary differential equations. A method is L-stable if it is A-stable and as , where is the stability function of the method. L-stable methods are in general very good at integrating stiff equations.
General linear methods (GLMs) are a large class of numerical methods used to obtain numerical solutions to ordinary differential equations. They include multistage Runge–Kutta methods that use intermediate collocation points, as well as linear multistep methods that save a finite time history of the solution. John C. Butcher originally coined this term for these methods, and has written a series of review papers a book chapter and a textbook on the topic. His collaborator, Zdzislaw Jackiewicz also has an extensive textbook on the topic. The original class of methods were originally proposed by Butcher (1965), Gear (1965) and Gragg and Stetter (1964).
Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. Because the linear part is integrated exactly, this can help to mitigate the stiffness of a differential equation. Exponential integrators can be constructed to be explicit or implicit for numerical ordinary differential equations or serve as the time integrator for numerical partial differential equations.
Hans Zanna Munthe-Kaas is a Norwegian mathematician at the University of Bergen, and UiT The Arctic University of Norway working in the area of computational mathematics in the borderland between pure and applied mathematics and computer science.
Sigal Gottlieb is an applied mathematician. She is a professor of mathematics and the director of the Center for Scientific Computing and Visualization Research at the University of Massachusetts Dartmouth.
Christian Lubich is an Austrian mathematician, specializing in numerical analysis.
Gerhard Wanner is an Austrian mathematician.
Doron Levy is a mathematician, scientist, magician, and academic. He is a Professor and chair at the Department of Mathematics at the University of Maryland, College Park. He is also the Director of the Brin Mathematics Research Center.
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