  We describe a NOVel form of Adaptive softening (NovA) for collisionless
$N$-body simulations, implemented in the Ramses adaptive mesh refinement code.
We introduce a refinement criterion that the particle distribution within each
cell be sufficiently isotropic, as measured by its moment of inertia tensor. In
this way, collapse is only refined if it occurs along all three axes, ensuring
that the softening $\epsilon$ is always of order twice the largest
inter-particle spacing in a cell. This more conservative force softening
criterion is designed to minimise spurious two-body effects, while maintaining
high force resolution in collapsed regions of the flow. We test NovA using an
antisymmetric perturbed plane wave collapse (`Valinia' test) before applying it
to warm dark matter (WDM) simulations. For the Valinia test, we show that --
unlike the standard $N$-body method -- NovA produces no numerical fragmentation
while still being able to correctly capture fine caustics and shells around the
collapsing regions. For the WDM simulations, we find that NovA converges
significantly more rapidly than standard $N$-body, producing little or no
spurious halos on small scales. We show, however, that determining whether or
not halos exist below the free streaming mass $M_{\rm fs}$ is complicated by
the fact that our halo finder (AHF) likely incorrectly labels some caustics and
criss-crossing filaments as halos, while one or two particularly massive
filaments appear to fragment in any version of NovA where refinement is
allowed. Such massive filaments may be physically unstable to collapse, as is
the case for infinite, static, self-gravitating cylinders. We will use NovA in
forthcoming papers to study the issue of halo formation below $M_{\rm fs}$;
filament stability; and to obtain new constraints on the temperature of dark
matter.
