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:

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 $h>\epsilon$.