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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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A history of MATLAB

Cleve B. Moler, +1 more
TL;DR: Enhancements to the MATLAB computing environment have dominated development in recent years, including extensions to the desktop, major enhancements to the object and graphics systems, support for parallel computing and GPUs, and the “Live Editor”, which combines programs, descriptive text, output and graphics into a single interactive, formatted document.
Journal ArticleDOI

Exponential time-differencing with embedded Runge-Kutta adaptive step control

Patrick Whalen, +2 more
- 01 Jan 2015 - 
TL;DR: Comparisons of performance with adaptive-stepping integrating factor are carried out on a set of canonical partial differential equations: the shock-fronts of Burgers equation, interacting KdV solitons, KS controlled chaos, and critical collapse of two-dimensional NLS.
Journal ArticleDOI

Applications of higher-order frequency response functions to the detection and damage assessment of general structural systems with breathing cracks

R.M. Lin, +1 more
- 01 Nov 2018 - 
TL;DR: A solid foundation has been laid for the potential applications of higher-order FRFs to damage identification and assessment of general real structural systems.
Journal ArticleDOI

Complex Flow Dynamics in Dense Granular Flows—Part II: Simulations

Piroz Zamankhan, +1 more
- 01 Jul 2007 - 
TL;DR: In this paper, a model was built to investigate the fluid dynamics of granular flows in an intermediate regime, where both collisional and frictional interactions may affect the flow behavior.
Proceedings ArticleDOI

Coupled Graph ODE for Learning Interacting System Dynamics

Zijie Huang, +2 more
TL;DR: In this paper, the authors proposed coupled graph ODE, a generative model that learns the coupled dynamics of nodes and edges with a graph neural network based ODE in a continuous manner.
References
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Book

Computational Methods in Ordinary Differential Equations

John Denholm Lambert
Journal ArticleDOI

Comparing Numerical Methods for Ordinary Differential Equations

T. E. Hull, +3 more
- 01 Dec 1972 - 
TL;DR: According to criteria involving the number of function evaluations, overhead cost, and reliability, the best general-purpose method, if function evaluations are not very costly, is one due to Bulirsch and Stoer, however, when function evaluated methods are relatively expensive, variable-order methods based on Adams formulas are best.

Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems

E. Fehlberg
Journal ArticleDOI

Coefficients for the study of Runge-Kutta integration processes

John C. Butcher
- 01 May 1963 - 
TL;DR: In this paper, a set of η first order simultaneous differential equations in the dependent variables y 1, y 2, y 3, y 4, y 5, y 6 and the independent variable x is considered.

Classical Fifth-, Sixth-, Seventh-, and Eighth-Order Runge-Kutta Formulas with Stepsize Control

E. Fehlberg
TL;DR: Runge-Kutta formulas of high order with stepsize control through leading truncation error term through leading parallelogram error term.