## 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:

$\begin{array}{ccc}\hfill {\tilde{y}}^{\text{'}}\left(t\right)& =& \tilde{f}\left(\tilde{y}\left(t\right)\right),\hfill \\ \hfill \tilde{y}\left(0\right)& =& {\tilde{y}}_{0},\hfill \end{array}$ | (13) |

on an invariant manifold $\tilde{\mathcal{M}}=\{y\in {\mathbb{R}}^{n};\tilde{g}\left(y\right)=0\}$, where $\tilde{f}$ and $\tilde{g}$ approximate $f$ and $g$
in a sense that remains to be defined.
The idea behind this approximation is to replace the differential
manifold $\mathcal{M}$ by a suitable approximation $\tilde{\mathcal{M}}$ for which an atlas consisting of *easily computable*
charts exists. If this is the case, one can reformulate the vector
field $\tilde{f}$ on each domain of the atlas in an *easy*
way. The main obstacle of *parametrization* methods
[56] or of *Lie-methods* [53] is then
overcome.

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

An obvious prerequisite for this idea to make sense is the existence of a neighborhood $\mathcal{V}$ of $\mathcal{M}$ containing the approximate manifold $\tilde{\mathcal{M}}$ 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 $\tilde{f}$ given $\tilde{g}$. By admissible, we mean tangent to the manifold $\tilde{\mathcal{M}}$, i.e. such that

where, for convenience, we have denoted $\tilde{G}\left(y\right)={\tilde{g}}^{\text{'}}\left(y\right)$. For any $y\in \tilde{\mathcal{M}}$, we can indeed define

where $P\left(y\right)={\tilde{G}}^{T}\left(y\right){\left(\tilde{G}\left(y\right){\tilde{G}}^{T}\left(y\right)\right)}^{-1}\tilde{G}\left(y\right)$ is the projection along $\tilde{\mathcal{M}}$.