## 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:

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

on an invariant manifold $\stackrel{˜}{ℳ}=\left\{y\in {ℝ}^{n};\stackrel{˜}{g}\left(y\right)=0\right\}$, where $\stackrel{˜}{f}$ and $\stackrel{˜}{g}$ approximate $f$ and $g$ in a sense that remains to be defined. The idea behind this approximation is to replace the differential manifold $ℳ$ by a suitable approximation $\stackrel{˜}{ℳ}$ for which an atlas consisting of easily computable charts exists. If this is the case, one can reformulate the vector field $\stackrel{˜}{f}$ 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 $\stackrel{˜}{ℳ}$. Nevertheless, it never drifts off the exact manifold considerably, if $ℳ$ and $\stackrel{˜}{ℳ}$ are chosen appropriately close to each other.

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

$\begin{array}{c}\hfill \forall \phantom{\rule{0.166667em}{0ex}}y\phantom{\rule{0.166667em}{0ex}}\in \phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{ℳ},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{G}\left(y\right)\stackrel{˜}{f}\left(y\right)=0,\end{array}$

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

 $\begin{array}{c}\hfill \stackrel{˜}{f}\left(y\right)=\left(I-P\left(y\right)\right)f\left(y\right),\end{array}$ (14)

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