Efficient Runge-Kutta methods for Hamiltonian equations
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Energy-preserving Runge-Kutta methods
Elena Celledoni, Robert I. McLachlan, David I. McLaren, Brynjulf Owren, G. Reinout W. Quispel, William M. Wright (2009)
ESAIM: Mathematical Modelling and Numerical Analysis
We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems.
Equilibria of Runge-Kutta methods.
E. Hairer, J.M. Sanz-Serna, A. Iserles (1990/1991)
Numerische Mathematik
Erratum: On the convergence of multistep methods for nonlinear stiff differential equations.
Lubich. C. (1992)
Numerische Mathematik
Error analysis of discrete approximations to bang-bang optimal control problems: the linear case
Vladimir Veliov (2005)
Control and Cybernetics
Estimation of longest stability interval for a kind of explicit linear multistep methods.
Xu, Y., Zhao, J.J. (2010)
Discrete Dynamics in Nature and Society
Euler scheme for SDEs with non-Lipschitz diffusion coefficient : strong convergence
Abdel Berkaoui, Mireille Bossy, Awa Diop (2008)
ESAIM: Probability and Statistics
We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form , . In that case, we study the rate of convergence of a symmetrized version of the Euler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz.
Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence
Abdel Berkaoui, Mireille Bossy, Awa Diop (2007)
ESAIM: Probability and Statistics
We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form |x|α, α ∈ [1/2,1). In that case, we study the rate of convergence of a symmetrized version of the Euler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz.
Explicit Runge-Kutta Formulas with Increased Stability Boundaries.
P.J. van der Houwen (1972/1973)
Numerische Mathematik
Explicit two-step Runge-Kutta methods
Zdzisław Jackiewicz, Rosemary Anne Renaut, Marino Zennaro (1995)
Applications of Mathematics
The explicit two-step Runge-Kutta (TSRK) formulas for the numerical solution of ordinary differential equations are analyzed. The order conditions are derived and the construction of such methods based on some simplifying assumptions is described. Order barriers are also presented. It turns out that for order the minimal number of stages for explicit TSRK method of order is equal to the minimal number of stages for explicit Runge-Kutta method of order . Numerical results are presented which...
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