arXiv subjects

Ander Murua

Publications and source records attributed to Ander Murua.

21 records · Page 2Linked to original sources

arXiv2010

Splitting methods with complex coefficients

Splitting methods for the numerical integration of differential equations of order greater than two involve necessarily negative coefficients. This order barrier can be overcome by considering complex coefficients with positive real part. In this work we review the composition technique used to construct methods of this class, propose new sixth-order integrators and analyze their main features on a pair of numerical examples, in particular how the errors are propagated along the evolution.

math.NA↗

arXiv2008

Splitting and composition methods in the numerical integration of differential equations

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 to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.

math.NA↗

arXiv2008

An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications

We provide a new algorithm for generating the Baker--Campbell--Hausdorff (BCH) series $Z = \log(\e^X \e^Y)$ in an arbitrary generalized Hall basis of the free Lie algebra $\mathcal{L}(X,Y)$ generated by $X$ and $Y$. It is based on the close relationship of $\mathcal{L}(X,Y)$ with a Lie algebraic structure of labeled rooted trees. With this algorithm, the computation of the BCH series up to degree 20 (111013 independent elements in $\mathcal{L}(X,Y)$) takes less than 15 minutes on a personal computer and requires 1.5 GBytes of memory. We also address the issue of the convergence of the series, providing an optimal convergence domain when $X$ and $Y$ are real or complex matrices.

math-ph↗