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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:

Im42 $\mtable{...}$(13)

on an invariant manifold Im43 ${\mover \#8499 \#732 ={{y\#8712 \#8477 ^n;\mover g\#732 {(y)}=0}}}$ , where Im44 $\mover f\#732 $ and Im45 $\mover g\#732 $ approximate f and g in a sense that remains to be defined. The idea behind this approximation is to replace the differential manifold Im22 $\#8499 $ by a suitable approximation Im46 $\mover \#8499 \#732 $ for which an atlas consisting of easily computable charts exists. If this is the case, one can reformulate the vector field Im44 $\mover f\#732 $ 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 Im46 $\mover \#8499 \#732 $ . Nevertheless, it never drifts off the exact manifold considerably, if Im22 $\#8499 $ and Im46 $\mover \#8499 \#732 $ are chosen appropriately close to each other.

An obvious prerequisite for this idea to make sense is the existence of a neighborhood Im47 $\#119985 $ of Im22 $\#8499 $ containing the approximate manifold Im46 $\mover \#8499 \#732 $ 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 Im44 $\mover f\#732 $ given Im45 $\mover g\#732 $ . By admissible, we mean tangent to the manifold Im46 $\mover \#8499 \#732 $ , i.e. such that

Im48 $\mtable{...}$

where, for convenience, we have denoted Im49 ${\mover G\#732 {(y)}=\mover g\#732 ^'{(y)}}$ . For any Im50 ${y\#8712 \mover \#8499 \#732 }$ , we can indeed define

Im51 $\mtable{...}$(14)

where Im52 ${P{(y)}=\mover G\#732 ^T{(y)}{(\mover G\#732 {(y)}\mover G\#732 ^T{(y)})}^{-1}\mover G\#732 {(y)}}$ is the projection along Im46 $\mover \#8499 \#732 $ .


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