Conservative closures of the Vlasov-Poisson equations based on symmetrically weighted Hermite spectral expansion

We derive conservative closures for the Vlasov-Poisson equations discretized in velocity via the symmetrically weighted Hermite spectral expansion. We demonstrate that no closure can simultaneously restore the conservation of mass, momentum, and energy in this formulation. The properties of the analytically derived conservative closures of each conserved quantity are validated numerically by simulating an electrostatic benchmark problem: the Langmuir wave. Both the numerical results and analytical analysis indicate that closure by truncation (i.e. setting the last Hermite moment to zero) is the most suitable conservative closure for the symmetrically weighted Hermite formulation.