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We present new symmetric fourth and sixth-order symplectic partitioned Runge–Kutta and Runge–Kutta–Nystr' om methods. We studied compositions using several extra stages, optimising the e1ciency. An e2ective error, Ef, is de3ned and an extensive search is carried out using the extra parameters. The new methods have smaller values of Ef than other methods… (More)

Approximate resolution of linear systems of differential equationswith varying coefficients is a recurrent problem, shared by a number of scientific and engineering areas, ranging from QuantumMechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to build up approximate exponential… (More)

We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple… (More)

- Sergio Blanes, Fernando Casas, Philippe Chartier, Ander Murua
- Math. Comput.
- 2013

We are concerned with the numerical solution obtained by splitting methods of certain parabolic partial differential equations. Splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients. It has been demonstrated that this second-order barrier can be overcome by using splitting methods with complex-valued… (More)

- Sergio Blanes
- 2000

In this paper we show how to build high order integrators for solving ordinary diierential equations by composition of low order methods and using the processing technique. From a basic p-th order method, p , one can obtain high order integrators in the processed form n = P K P ?1 (n > p) being both the processor P and the kernel K compositions of the basic… (More)

- Sergio Blanes, Fasma Diele, Carmela Marangi, Stefania Ragni
- J. Computational Applied Mathematics
- 2010

- S. Blanes, P. C. Moan
- 2001

We present an algorithm for numerically integrating non-autonomous Hamiltonian differential equations. Special attention is paid to the separable case and, in particular, a new fourth-order splitting method is presented which in a certain measure is optimal. In combination with a new way of handling non-autonomous problems, the schemes we present are based… (More)

- Sergio Blanes, Fernando Casas
- Numerical Algorithms
- 2004

We analyse composition and polynomial extrapolation as procedures to raise the order of a geometric integrator for solving numerically differential equations. Methods up to order sixteen are constructed starting with basic symmetric schemes of order six and eight. If these are geometric integrators, then the new methods obtained by extrapolation preserve… (More)

We present new symmetric fourth and sixth-order symplectic Partitioned Runge{ Kutta and Runge{Kutta{Nystrr om methods. We studied compositions using several extra stages, optimising the eeciency. An eeective error, E f , is deened and an extensive search is carried out using the extra parameters. The new methods have smaller values of E f than other methods… (More)

- Sergio Blanes, Fernando Casas, José Ros
- SIAM J. Scientific Computing
- 1999