  Dust grains play a significant role in several astrophysical processes,
including gas/dust dynamics, chemical reactions, and radiative transfer.
Replenishment of small-grain populations is mainly governed by fragmentation
during pair-wise collisions between grains. The wide spectrum of fragmentation
outcomes, from complete disruption to erosion and/or mass transfer, can be
modelled by the general non-linear fragmentation equation. Efficiently solving
this equation is crucial for an accurate treatment of the dust fragmentation in
numerical modelling. However, similar to dust coagulation, numerical errors in
current fragmentation algorithms employed in astrophysics are dominated by the
numerical over-diffusion problem -- particularly in 3D hydrodynamic simulations
where the discrete resolution of the mass density distribution tends to be
highly limited. With this in mind, we have derived the first conservative form
of the general non-linear fragmentation with a mass flux highlighting the mass
transfer phenomenon. Then, to address cases of limited mass density resolution,
we applied a high-order discontinuous Galerkin scheme to efficiently solve the
conservative fragmentation equation with a reduced number of dust bins. An
accuracy of 0.1 -1% is reached with 20 dust bins spanning a mass range of 9
orders of magnitude.
