  We study the asymptotic behaviour of the following linear
growth-fragmentation equation$$\dfrac{\partial}{\partial t} u(t,x) +
\dfrac{\partial}{\partial x} \big(x u(t,x)\big) + B(x) u(t,x) =4
B(2x)u(t,2x),$$ and prove that under fairly general assumptions on the division
rate $B(x),$ its solution converges towards an oscillatory function,explicitely
given by the projection of the initial state on the space generated by the
countable set of the dominant eigenvectors of the operator. Despite the lack of
hypo-coercivity of the operator, the proof relies on a general relative entropy
argument in a convenient weighted $L^2$ space, where well-posedness is obtained
via semigroup analysis. We also propose a non-dissipative numerical scheme,
able to capture the oscillations.
