A family of embedded Runge-Kutta formulae
J. R. Dormand,P.J. Prince +1 more
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.read more
Citations
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High order computationally economical six-step method with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation
TL;DR: A computationally economical symmetric six-step algorithm with high algebraic and phase-lag order is obtained in this article. But the complexity of the algorithm is not analyzed in this paper.
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An Iterative Variable-timestep Algorithm for Molecular Dynamics Simulations
TL;DR: The variable-timestep method was developed specifically for simulations involving disparate timescales, such as are encountered with the reactive bond-order model used here, but it should also prove beneficial for a wide range of molecular dynamics applications.
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Dynamical chaos for a limited power supply for fluid oscillations in cylindrical tanks
TL;DR: Forced oscillations of the fluid surface in a cylindrical tank due to interaction with the excitation mechanism of a limited power supply (so-called limited excitation) are investigated in detail in this article.
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A new multistep method with optimized characteristics for initial and/or boundary value problems
TL;DR: In this article, an optimized multistage symmetric two-step method is proposed for the first time in the literature, which is considered as optimized due to the following reasons: (1) it is of tenth-algebraic order scheme, (2) it obliterated the phase-lag and its first, second, third and fourth derivatives, (3) it improved stability characteristics, (4) it was a P-stable method.
Particle Smoothing in Continuous Time: A Fast Approach via Density Estimation
TL;DR: A pair of variants using kernel density approximations to relieve the dependence of particle smoothing for continuous-time, nonlinear models expressed via stochastic differential equations (SDEs) demonstrate comparable accuracy and significantly improved runtime over conventional techniques.
References
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Comparing Numerical Methods for Ordinary Differential Equations
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.
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Coefficients for the study of Runge-Kutta integration processes
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
TL;DR: Runge-Kutta formulas of high order with stepsize control through leading truncation error term through leading parallelogram error term.
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