Anti-symmetric and Positivity Preserving Formulation of a Spectral Method for Vlasov-Poisson Equations

Abstract

We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

Figures (14)

• Figure 1: The (a) SW and (b) SW square-root formulations manufactured solution distribution function $f^{m}(x, v, t=1)$ results with $N_{x} = N_{v}=100$. Subfigures (c) and (d) show the relative error between the analytic solution and the SW and SW square-root numerical results, respectively. The numerical results show that as expected the SW and SW square-root formulation accuracy is comparable, yet the SW formulation has more interaction between higher-order modes.
• Figure 2: Subfigure (a) shows the $L^2$-norm of the expansion coefficients with $N_{v}=N_{x}=100$. The spectral convergence illustrates that the SW square-root formulation requires fewer spectral coefficients in comparison to the SW formulation to approximate the manufactured solution. Subfigure (b) shows the $L^2$-norm absolute error of the SW and SW square-root manufactured solutions. In subfigure (b), we set $N_{v}=100$ and $N_{x} = 50, 100, 200, 400$. The numerical convergence rate is $2$ for both formulations, which agrees with the second-order central finite difference convergence rate.
• Figure 3: Conservation of particle number, momentum, and energy via the SW formulation (a/b) and the SW square-root formulation (c/d) for linear Landau damping. Subfigures (a/c) have an odd number of spectral terms ($N_v = 101$) and subfigures (b/d) have an even number of spectral terms ($N_v = 100$). The numerical drift rate (the rate at which conservation no longer holds) in each setup matches the analytic drift rate, which we derive in Eqns. \ref{['change-in-mass-sw']}, \ref{['change-in-momentum-sw']}, \ref{['change-in-total-energy-sw']}, \ref{['drift-in-momentum-sw-sqrt']}, and \ref{['change-in-total-energy-sw-sqrt']}.
• Figure 4: The linear Landau damping electric field amplitude as a function of time computed via the SW square-root formulation. The numerical damping rate agrees with the theoretical damping rate from landau_damping_rate_1973. The forward and backward simulations are evolved via the non-adaptive 3rd-order explicit Runge-Kutta integrator of Bogacki-Shampine bogacki_shampine_1989_integrator with $\Delta t = 10^{-3}$ and $N_{v}=100$. Note that such an explicit Runge-Kutta temporal integrator is not time-reversal symmetric, yet the anti-symmetric structure of the equations makes this method approximately time-reversible.
• Figure 5: The evolution of the electron distribution function $f^{e}(x, v, t)$ for nonlinear Landau damping with $N_{v}=100$. The distribution function becomes negative in the (a) SW formulation whereas the (b) SW square-root formulation is positivity preserving by construction.

Theorems & Definitions (19)

• Proposition 1
• Definition 1: Adjoint of an operator renardy_2004_introduction
• Example 1
• Example 2
• Definition 2: Anti-symmetric operator halpern_2021_antisymmetric
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Remark 2