Not yet reviewed by Pith; the record is open.

Recently, we have shown how current cosmological N-body codes already follow the fine grained phase-space information of the dark matter fluid. Using a tetrahedral tesselation of the three-dimensional manifold that describes perfectly cold fluids in six-dimensional phase space, the phase-space distribution function can be followed throughout the simulation. This allows one to project the distribution function into configuration space to obtain highly accurate densities, velocities, and velocity dispersions. Here, we exploit this technique to show first steps on how to devise an improved particle-mesh technique. At its heart, the new method thus relies on a piecewise linear approximation of the phase space distribution function rather than the usual particle discretisation. We use pseudo-particles that approximate the masses of the tetrahedral cells up to quadrupolar order as the locations for cloud-in-cell (CIC) deposit instead of the particle locations themselves as in standard CIC deposit. We demonstrate that this modification already gives much improved stability and more accurate dynamics of the collisionless dark matter fluid at high force and low mass resolution. We demonstrate the validity and advantages of this method with various test problems as well as hot/warm-dark matter simulations which have been known to exhibit artificial fragmentation. This completely unphysical behaviour is much reduced in the new approach. The current limitations of our approach are discussed in detail and future improvements are outlined.