Splitting and composition methods in the numerical integration of differential equations

Details Splitting and composition methods in the numerical integration of differential equations Splitting and composition methods in the numerical integration of differential equations

2008-12-01

arxiv.org/abs/0812.0377v1

Bol. Soc. Esp. Mat. Apl. 45 (2008), 89-145

Mathematics

Numerical Analysis

Review paper; 56 pages, 6 figures, 8 tables

Scientific paper

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.

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