  This paper explores energy-, momentum-, density-, and positivity-preserving
spatio-temporal discretizations for the nonlinear Landau collision operator. We
discuss two approaches, namely direct Galerkin formulations and discretizations
of the underlying infinite-dimensional metriplectic structure of the collision
integral. The spatial discretizations are chosen to reproduce the
time-continuous conservation laws that correspond to Casimir invariants and to
guarantee the positivity of the distribution function. Both the direct and the
metriplectic discretization are demonstrated to have exact H-theorems and
unique, physically exact equilibrium states. Most importantly, the two
approaches are shown to coincide, given the chosen Galerkin method. A temporal
discretization, preserving all of the mentioned properties, is achieved with
so-called discrete gradients. Hence the proposed algorithm successfully
translates all properties of the infinite-dimensional time-continuous Landau
collision operator to time- and space-discrete sparse-matrix equations suitable
for numerical simulation.
