### SPLITTING AND COMPOSITION METHODS IN THE NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS

*S. Blanes, F. Casas, A. Murua*

#### Resumen

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.

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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