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Open AccessJournal ArticleDOI

A family of embedded Runge-Kutta formulae

J. R. Dormand, +1 more
- 01 Mar 1980 - 
- Vol. 6, Iss: 1, pp 19-26
TLDR
In this article, a family of embedded Runge-Kutta formulae RK5 (4) are derived from these and a small principal truncation term in the fifth order and extended regions of absolute stability.
About: 
This article is published in Journal of Computational and Applied Mathematics.The article was published on 1980-03-01 and is currently open access. It has received 3106 citations till now.

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Citations
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Journal ArticleDOI

The MATLAB ODE Suite

Lawrence F. Shampine, +1 more
- 01 Jan 1997 - 
TL;DR: This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in MATLAB.
Book

Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-dependent Problems

Randall J. LeVeque
TL;DR: This book discusses infinite difference approximations, Iterative methods for sparse linear systems, and zero-stability and convergence for initial value problems for ordinary differential equations.
Posted Content

Score-Based Generative Modeling through Stochastic Differential Equations

Yang Song, +5 more
- 26 Nov 2020 - 
TL;DR: This work presents a stochastic differential equation (SDE) that smoothly transforms a complex data distribution to a known prior distribution by slowly injecting noise, and a corresponding reverse-time SDE that transforms the prior distribution back into the data distribution by Slowly removing the noise.
MonographDOI

Solving ODEs with MATLAB

Ian Gladwell, +2 more
TL;DR: In this article, the authors provide a sound treatment of ODEs with Matlab in about 250 pages, with a discussion of "the facts of life" for the problem, mainly by means of examples.
Journal ArticleDOI

A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides

Jeff Cash, +1 more
- 01 Sep 1990 - 
TL;DR: A family of explicit Runge-Kutta formulas that contains imbedded formulas of all orders 1 through 4 is derived, which is very efficient for problems with smooth solution as well as problems having rapidly varying solutions.
References
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Journal ArticleDOI

An Order Five Runge-Kutta Process with Extended Region of Stability

J. Douglas Lawson
- 01 Dec 1966 - 

Classical eight- and lower-order Runge-Kutta-Nystroem formulas with stepsize control for special second-order differential equations

E. Fehlberg
TL;DR: The formulas discussed save 50 percent or more computer time compared with other Runge-Kutta-Nystrom formulas if the latter are operated by using the standard procedure for stepsize control.
Journal ArticleDOI

A Theoretical Criterion for Comparing Runge–Kutta Formulas

Kenneth R. Jackson, +2 more
- 01 Jun 1978 - 
TL;DR: In this article, a criterion is proposed for determining which explicit Runge-Kutta formulas are the most promising as a basis for developing good library subroutines for solving nonstiff initial-value problems associated with ordinary differential equations.
Journal ArticleDOI

A-stable one-step methods with step-size control for stiff systems of ordinary differential equations

René Alt
- 01 Jan 1978 - 
TL;DR: Two efficient third-and fourth-order processes for solving the initial value problem for ordinary differential equations are studied and both are A-stable and so recommended for stiff systems.
Journal ArticleDOI

Quadrature and Runge-Kutta formulas

L. F. Shampire
- 01 Jan 1976 - 
TL;DR: Error estimation and step size adjustment procedures for many high order Runge-Kutta methods fail dramatically when applied to an equation of the form y'(x)=f(x), and a practical remedy is proposed and its use exemplified with the Fehlberg [7,8] formulas as discussed by the authors.