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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 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.
John Charles Butcher is a New Zealand mathematician who specialises in numerical methods for the solution of ordinary differential equations.
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 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.
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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.
The Segregated Runge–Kutta (SRK) method is a family of IMplicit–EXplicit (IMEX) Runge–Kutta methods that were developed to approximate the solution of differential algebraic equations (DAE) of index 2.
A Lie group integrator is a numerical integration method for differential equations built from coordinate-independent operations such as Lie group actions on a manifold. They have been used for the animation and control of vehicles in computer graphics and control systems/artificial intelligence research. These tasks are particularly difficult because they feature nonholonomic constraints.
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.
Antonella Zanna Munthe-Kaas is an Italian applied mathematician and numerical analyst whose research includes work on numerical integration of differential equations and applications to medical imaging. She is a professor and head of the mathematics department at the University of Bergen in Norway.
Gerhard Wanner is an Austrian mathematician.
Assyr Abdulle was a Swiss mathematician. He specialized in numerical mathematics.
References
- ↑ "Hans Munthe-Kaas - The Mathematics Genealogy Project". genealogy.math.ndsu.nodak.edu. Retrieved 16 March 2017.
- ↑ "Research". Hans.Munthe-Kaas.no. Retrieved 16 June 2017.
- ↑ Munthe-Kaas, Hans (1995). "Lie-Butcher Therory for Runge-Kutta Methods" (PDF). BIT Numerical Mathematics. 35 (4): 572–587. doi:10.1007/BF01739828. S2CID 16255510.
- ↑ Munthe-Kaas, Hans (1998). "Runge-Kutta methods on Lie groups" (PDF). BIT Numerical Mathematics. 38: 92–111. doi:10.1007/BF02510919. S2CID 14427246.
- ↑ Munthe-Kaas, Hans (January 1999). "High order Runge-Kutta methods on manifolds" (PDF). Applied Numerical Mathematics. 29 (1): 115–127. doi:10.1016/S0168-9274(98)00030-0.
- ↑ Iserles, Arieh; Munthe-Kaas, Hans; Nørsett, Syvert P.; Zanna, Antonella. "Lie-group methods". Acta Numerica. 2000: 215–365.
- ↑ "Short curriculum vitae: Hans Munthe-Kaas" . Retrieved 2020-04-06.
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