## Section: Scientific Foundations

### Highly-oscillatory systems

Participants : François Castella, Philippe Chartier, Erwan Faou.

In applications to molecular dynamics or quantum dynamics for instance, the right-hand side of (1 ) involves *fast* forces (short-range interactions) and *slow* forces (long-range interactions). Since *fast* forces are much cheaper to evaluate than *slow* forces, it seems highly desirable to design numerical methods for which the number of evaluations of slow forces is not (at least not too much) affected by the presence of fast forces.

A typical model of highly-oscillatory systems is the second-order differential equations

where the potential $V\left(q\right)$ is a sum of potentials $V=W+U$ acting on different time-scales, with ${\nabla}^{2}W$ positive definite and $\parallel {\nabla}^{2}W\parallel >>\parallel {\nabla}^{2}U\parallel $. In order to get a bounded error propagation in the linearized equations for an explicit numerical method, the step size must be restricted according to

where $C$ is a constant depending on the numerical method and where $\omega $ is the highest frequency of the problem, i.e. in this situation the square root of the largest eigenvalue of ${\nabla}^{2}W$. In applications to molecular dynamics for instance, *fast* forces deriving from $W$ (short-range interactions) are much cheaper to evaluate than *slow* forces deriving from $U$ (long-range interactions). In this case, it thus seems highly desirable to design numerical methods for which the number of evaluations of slow forces is not (at least not too much) affected by the presence of fast forces.

Another prominent example of highly-oscillatory systems is encountered in quantum dynamics where the Schrödinger equation is the model to be used. Assuming that the Laplacian has been discretized in space, one indeed gets the *time*-dependent Schrödinger equation:

$\begin{array}{c}\hfill i\dot{\psi}\left(t\right)=\frac{1}{\epsilon}H\left(t\right)\psi \left(t\right),\end{array}$ | (7) |

where $H\left(t\right)$ is finite-dimensional matrix and where $\epsilon $ typically is the square-root of a mass-ratio (say electron/ion for instance) and is small ($\epsilon \approx {10}^{-2}$ or smaller). Through the coupling with classical mechanics ($H\left(t\right)$ is obtained by solving some equations from classical mechanics), we are faced once again with two different time-scales, 1 and $\epsilon $. In this situation also, it is thus desirable to devise a numerical method able to advance the solution by a time-step $h>\epsilon $.