A conditional formulation of the Vlasov-Ampere Equations: A conservative, positivity, asymptotic, and Gauss law preserving scheme

We propose a novel reformulation of the Vlasov-Amp{\`e}re equations for plasmas. This reformulation exposes discrete symmetries to achieve simultaneous conservation of mass, momentum, and energy; preservation of Gauss's law involution; positivity of the distribution function; and quasi-neutral asymptotics. Our approach relies on transforming variables and coordinates, leading to a coupled system of a modified Vlasov equation and the associated moment-field equations. The modified Vlasov equation evolves a conditional distribution function that excludes information on mass, momentum, and energy densities. The mass, momentum, and energy density, in turn, are evolved using moment equations, in which discrete symmetries, conservation laws, and involution constraints are enforced. The reformulation is compatible with a recent slow-manifold reduction technique, which separates the fast electron time scales and simplifies handling the asymptotic quasi-neutrality limit within the easier to solve moment-field subsystem. Using this new formulation, we develop a numerical approach for the reduced 1D1V subsystem that, for the first time, simultaneously satisfies the important physical constraints while preserving the quasi-neutrality asymptotic limit. The advantages of our approach are demonstrated through canonical electrostatic test problems, including the multiscale ion acoustic shock wave problem.