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THE DYNAMICS OF RUNGE–KUTTA METHODS
International Journal of Bifurcation and Chaos, 1992The first step in investigating the dynamics of a continuous-time system described by an ordinary differential equation is to integrate to obtain trajectories. In this paper, we attempt to elucidate the dynamics of the most commonly used family of numerical integration schemes, Runge–Kutta methods, by the application of the techniques of dynamical ...
Cartwright, Julyan H. E., Piro, Oreste
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Improved Runge–Kutta–Chebyshev methods
Mathematics and Computers in Simulation, 2020zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Xiao Tang, Aiguo Xiao
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On the Stability of Volterra–Runge–Kutta Methods
SIAM Journal on Numerical Analysis, 1984This paper examines the stability properties of extended Runge-Kutta methods when applied to Volterra integral equations of the second kind of the form \[ y(x)=f(x)+\lambda \int^{x}_{0}k(x-s)y(s)ds\quad(x\geq 0) \] where \(Re(\lambda)0\), \(k_ 0(x)=\overline{k_ 0(-x)}\), \(x\leq 0\) there is no such order barrier.
Hairer, Ernst, Lubich, Christian
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Interpolation for Runge–Kutta Methods
SIAM Journal on Numerical Analysis, 1985The author discusses a new method for interpolation between mesh points of Runge-Kutta algorithms for the approximate solution of ordinary differential equations. The method is shown to fall under the classification of scaled Runge-Kutta algorithms as considered by \textit{M. K. Horn} [ibid. 20, 558-568 (1983; Zbl 0511.65048)].
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Multiplicative Runge–Kutta methods
Nonlinear Dynamics, 2007zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Efficient symplectic Runge–Kutta methods
Applied Mathematics and Computation, 2006The authors consider the efficiency of symplectic Runge-Kutta methods with real eigenvalues for the numerical integration of initial value problems for systems of ordinary differential equations.
Robert P. K. Chan +2 more
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2019
The class of differential equations for which explicit solutions can be obtained is rather small. In fact, in Chap. 3, we have already remarked that to find an explicit solution of the second-order linear differential equation ( 3.2) there does not exist any method.
Ravi P. Agarwal +2 more
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The class of differential equations for which explicit solutions can be obtained is rather small. In fact, in Chap. 3, we have already remarked that to find an explicit solution of the second-order linear differential equation ( 3.2) there does not exist any method.
Ravi P. Agarwal +2 more
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Positivity of Runge-Kutta and diagonally split Runge-Kutta methods
Applied Numerical Mathematics, 1998The author investigates positivity of general Runge-Kutta and diagonally split Runge-Kutta methods for the numerical solution of positive initial value problems for ordinary differential equations. Conditions for the maximal stepsize in term of the radius of positivity of the Runge-Kutta method which guarantees positivity are given.
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A Note on a Runge‐Kutta‐Chebyshev Method
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1982Le but de ce travail est de présenter quelques considérations sur la solution numérique d'un problème avec valeur initiale pour des systèmes d'équations différentielles ordinaires de la forme \(y'=f(y)\), qui jouïssent de la propriété que les valeurs propres de la matrice jacobienne \(J(f)=\partial f(y)/\partial y\) sont situées sur une bande longue et
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Runge–Kutta–Chebyshev projection method
Journal of Computational Physics, 2006In this paper a fully explicit, stabilized projection method called the Runge-Kutta-Chebyshev (RKC) projection method is presented for the solution of incompressible Navier-Stokes systems. This method preserves the extended stability property of the RKC method for solving ODEs, and it requires only one projection per step.
Zheming Zheng, Linda R. Petzold
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