  The micro-macro (mM) decomposition approach is considered for the numerical
solution of the Vlasov--Poisson--Lenard--Bernstein (VPLB) system, which is
relevant for plasma physics applications. In the mM approach, the kinetic
distribution function is decomposed as
$f=\mathcal{E}[\boldsymbol{\rho}_{f}]+g$, where $\mathcal{E}$ is a local
equilibrium distribution, depending on the macroscopic moments
$\boldsymbol{\rho}_{f}=\int_{\mathbb{R}}\boldsymbol{e}
fdv=\langle\boldsymbol{e} f\rangle_{\mathbb{R}}$, where
$\boldsymbol{e}=(1,v,\frac{1}{2}v^{2})^{\rm{T}}$, and $g$, the microscopic
distribution, is defined such that $\langle\boldsymbol{e}
g\rangle_{\mathbb{R}}=0$. We aim to design numerical methods for the mM
decomposition of the VPLB system, which consists of coupled equations for
$\boldsymbol{\rho}_{f}$ and $g$. To this end, we use the discontinuous Galerkin
(DG) method for phase-space discretization, and implicit-explicit (IMEX) time
integration, where the phase-space advection terms are integrated explicitly
and the collision operator is integrated implicitly. We give special
consideration to ensure that the resulting mM method maintains the
$\langle\boldsymbol{e} g\rangle_{\mathbb{R}}=0$ constraint, which may be
necessary for obtaining (i) satisfactory results in the collision dominated
regime with coarse velocity resolution, and (ii) unambiguous conservation
properties. The constraint-preserving property is achieved through a consistent
discretization of the equations governing the micro and macro components. We
present numerical results that demonstrate the performance of the mM method.
The mM method is also compared against a corresponding DG-IMEX method solving
directly for $f$.
