Journal ArticleDOI
Third-order composite Runge–Kutta method for stiff problems
TLDR
Combination of the harmonic and arithmetic means of the Runge–Kutta formulation has resulted in the introduction of a new formula for the numerical solution of stiff ordinary differential equations.Abstract:
The derivation of a composite method for solving stiff ordinary differential equations is discussed. Combination of the harmonic and arithmetic means of the Runge–Kutta formulation has resulted in the introduction of a new formula for the numerical solution of stiff ordinary differential equations. The numerical results and the A-stability of this new formula are examined.read more
Citations
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Journal ArticleDOI
A new variable step size block backward differentiation formula for solving stiff initial value problems
TL;DR: A new block backward differentiation formula of order 4 with variable step size is formulated, by varying a parameter in the formula, different sets of formulae with A-stability property can be generated.
Proceedings Article
Fifth-order mean Runge-Kutta methods applied to the Lorenz system
TL;DR: In this paper, the fifth-order Runge-Kutta methods were applied to the Lorenz system and compared with another method using harmonic mean with the same order, making comparisons made between two time steps and the accuracy of all methods from a non-chaotic to a chaotic system.
Journal ArticleDOI
A Comparison of Explicit Semi-Analytical Numerical Integration Methods for Solving Stiff ODE Systems
TL;DR: A comparison among three semi-analytical numerical integration algorithms for solving stiff ODE systems is presented and the accuracy, stability, computational efficiency and error and stability analysis of each method are presented.
Journal ArticleDOI
Solving stiff differential equations using a-stable block method
TL;DR: In this article, a two-point block one-step method for solving stiff ODEs is presented, where the solutions will be obtained simultaneously in block and produces two approximate solutions using constant step size.
References
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Book
Mathematica: A System for Doing Mathematics by Computer
TL;DR: This book will be released simultaneously with Release 2.0 of Mathematica and will cover all the new features of Release 1.0 as mentioned in this paper, including 16 pages of full-color graphics.
Book
Mathematica: a system for doing mathematics by computer (2nd ed.)
TL;DR: This new edition maintains the format of the original book and is the single most important user guide and reference for Mathematica--all users ofMathematica will need this edition.
Journal ArticleDOI
Numerical methods for ordinary differential equations in the 20th century
TL;DR: This survey paper follows the main strands in the developments of these methods, both for general problems, stiff systems, and for many of the special problem types that have been gaining in significance as the century draws to an end.
Journal ArticleDOI
A review of recent developments in solving ODEs
TL;DR: This review looks at techniques for dealing with special problems that may arise in some ODEs, for example, discontinuities, and discusses stiff equations, why they are difficult to solve, and methods and software for solving both nonstiff and stiff equations.
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