## Section: Scientific Foundations

### Spatial approximation for solving ODEs

Participants : Philippe Chartier, Erwan Faou.

The technique consists in solving an approximate initial value problem
on an approximate invariant manifold for which an atlas consisting of *easily computable* charts exists. The numerical solution obtained is this way never drifts off the exact manifold considerably even for long-time integration.

Instead of solving the initial Cauchy problem, the technique consists in solving an approximate initial value problem of the form:

on an invariant manifold , where and approximate f and g
in a sense that remains to be defined.
The idea behind this approximation is to replace the differential
manifold by a suitable approximation for which an atlas consisting of *easily computable*
charts exists. If this is the case, one can reformulate the vector
field on each domain of the atlas in an *easy*
way. The main obstacle of *parametrization* methods
[51] or of *Lie-methods* [47] is then
overcome.

The numerical solution obtained is this way obviously does not lie
on the exact manifold: it lives on the approximate manifold
. Nevertheless, it never drifts off the exact
manifold considerably, if and are
chosen appropriately *close* to each other.

An obvious prerequisite for this idea to make sense is the existence of a neighborhood of containing the approximate manifold and on which the vector field f is well-defined. In contrast, if this assumption is fulfilled, then it is possible to construct a new admissible vector field given . By admissible, we mean tangent to the manifold , i.e. such that

where, for convenience, we have denoted . For any , we can indeed define

where is the projection along .