  To study the resolution required for simulating gravitational fragmentation
with newly developed Lagrangian hydrodynamic schemes, Meshless Finite Volume
method (MFV) and Meshless Finite Mass method (MFM), we have performed a number
of simulations of the Jeans test and compared the results with both the
expected analytic solution and results from the more standard Lagrangian
approach: Smoothed Particle Hydrodynamics (SPH). We find that the different
schemes converge to the analytic solution when the diameter of a fluid element
is smaller than a quarter of the Jeans wavelength, $\lambda_\mathrm{J}$. Among
the three schemes, SPH/MFV shows the fastest/slowest convergence to the
analytic solution. Unlike the well-known behaviour of Eulerian schemes, none of
the Lagrangian schemes investigated displays artificial fragmentation when the
perturbation wavelength, $\lambda$, is shorter than $\lambda_\mathrm{J}$, even
at low numerical resolution. For larger wavelengths ($\lambda >
\lambda_\mathrm{J}$) the growth of the perturbation is delayed when it is not
well resolved. Furthermore, with poor resolution, the fragmentation seen with
the MFV scheme proceeds very differently compared to the converged solution.
All these results suggest that, when unresolved, the ratio of the magnitude of
hydrodynamic force to that of self-gravity at the sub-resolution scale is the
largest/smallest in MFV/SPH, the reasons for which we discussed in detail.
These tests are repeated to investigate the effect of kernels of higher-order
than the fiducial cubic spline. Our results indicate that the standard
deviation of the kernel is a more appropriate definition of the size of a fluid
element than its compact support radius.
