  "Generalized Hydrodynamics" (GHD) stands for a model that describes
one-dimensional \textit{integrable} systems in quantum physics, such as
ultra-cold atoms or spin chains. Mathematically, GHD corresponds to nonlinear
equations of kinetic type, where the main unknown, a statistical distribution
function $f(t,z,\theta)$, lives in a phase space which is constituted by a
one-dimensional position variable $z$, and a one-dimensional "kinetic" variable
$\theta$, actually a wave-vector, called "rapidity". Two key features of GHD
equations are first a non-local and nonlinear coupling in the advection term,
and second an infinite set of conserved quantities, which prevent the system
from thermalizing. To go beyond this, we consider the dissipative GHD
equations, which are obtained by supplementing the right-hand side of the GHD
equations with a non-local and nonlinear diffusion operator or a Boltzmann-type
collision integral. In this paper, we deal with new high-order numerical
methods to efficiently solve these kinetic equations. In particular, we devise
novel backward semi-Lagrangian methods for solving the advective part (the
so-called Vlasov equation) by using a high-order time-Taylor series expansion
for the advection fields, whose successive time derivatives are obtained by a
recursive procedure. This high-order temporal approximation of the advection
fields are used to design new implicit/explicit Runge-Kutta semi-Lagrangian
methods, which are compared to Adams-Moulton semi-Lagrangian schemes. For
solving the source terms, constituted by the diffusion and collision operators,
we use and compare different numerical methods of the literature.
