  Study of far-from-equilibrium thermalization dynamics in quantum materials,
including the dynamics of different types of quasiparticles, is becoming
increasingly crucial. However, the inherent complexity of either the full
quantum mechanical treatment or the solution of the scattering integral in the
Boltzmann approach, has significantly limited the progress in this domain. In
our previous work we had developed a solver to calculate the scattering
integral in the Boltzmann equation. The solver is free of any approximation (no
linearisation of the scattering operator, no close-to-equilibrium
approximation, full non-analytic dispersions, full account of Pauli factors,
and no limit to low order scattering) \cite{Michael}. Here we extend it to
achieve a higher order momentum space convergence by extending to second degree
basis functions.cWe further use an adaptive time stepper, achieving a
significant improvement in the numerical performance. Moreover we show adaptive
time stepping can prevent intrinsic instabilities in the time propagation of
the Boltzmann scattering operator. This work makes the numerical time
propagation of the full Boltzmann scattering operator efficient, stable and
minimally reliant on human supervision.
