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

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 eeciency. An eeective error, E f , is deÿned and an extensive search is carried out using the extra parameters. The new methods have smaller values of E f than other methods… (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, Ander Murua
- The Journal of chemical physics
- 2006

We present a family of symplectic splitting methods especially tailored to solve numerically the time-dependent Schrodinger equation. When discretized in time, this equation can be recast in the form of a classical Hamiltonian system with a Hamiltonian function corresponding to a generalized high-dimensional separable harmonic oscillator. The structure of… (More)

- Sergio Blanes, Fernando Casas, Ander Murua
- Foundations of Computational Mathematics
- 2008

A comprehensive linear stability analysis of splitting methods is carried out by means of a 2 × 2 matrix K(x) with polynomial entries (the stability matrix) and the stability polynomial p(x) (the trace of K(x) divided by two). An algorithm is provided for determining the coefficients of all possible time-reversible splitting schemes for a prescribed… (More)

- Sergio Blanes, Fernando Casas, Ander Murua
- SIAM J. Scientific Computing
- 2011

- 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, Fernando Casas, Jesús María Sanz Serna
- SIAM J. Scientific Computing
- 2014

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

- 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)