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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 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)

In this paper new integration algorithms for linear diierential equations up to eighth order are obtained. Starting from Magnus expansion, methods based on Cayley transformation and Fer expansion are also built. The structure of the exact solution is retained while the computational cost is reduced compared to similar methods. Their relative performance is… (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)

We build high order eecient numerical integration methods for solving the linear diierential equation _ X = A(t)X based on Magnus expansion. These methods preserve qualitative geometric properties of the exact solution and involve the use of single integrals and fewer commutators than previously published schemes. Sixth-and eighth-order numerical algorithms… (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, 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)

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