Conservative DG Method for the Micro-Macro Decomposition of the Vlasov-Poisson-Lenard-Bernstein Model

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 $f$.