Boletín SEMA, No 45 (2008)

Tamaño de la letra:  Pequeña  Mediana  Grande

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.

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