On the quadratic stability of asymmetric Hermite basis with application to plasma physics with oscillating electric field

TL;DR

This work analyzes why asymmetric Hermite-based discretizations of transport lose quadratic stability after truncation, due to the loss of skew-symmetry with respect to the Gram matrix $A$. It proves exact formulas for the scalar product of asymmetric Hermite functions and introduces two stabilization strategies: a low-rank correction that restores $A^N\overline D^N+(\overline D^N)^T A^N=0$ and a penalization approach that preserves a penalized quadratic energy; these methods generalize to other transport equations and to split-operator Vlasov-type systems. Numerical experiments with oscillating electric fields demonstrate unconditional stability and robustness for the proposed schemes, including applications to the diocotron and two-stream instabilities. The results offer practical, energy-consistent discretizations for plasma-transport models that were previously prone to instability with asymmetric bases.

Abstract

We analyze why the discretization of linear transport with asymmetric Hermite basis functions can be instable in quadratic norm. The main reason is that the finite truncation of the infinite moment linear system looses the skew-symmetry property with respect to the Gram matrix. Then we propose an original closed formula for the scalar product of any pair of asymmetric basis functions. It makes possible the construction of two simple modifications of the linear systems which recover the skew-symmetry property. By construction the new methods are quadratically stable with respect to the natural $L^2$ norm. We explain how to generalize to other transport equations encountered in numerical plasma physics. Basic numerical tests with oscillating electric fields of different nature illustrate the unconditional stability properties of our algorithms.

Figures (6)

• Figure 1: Numerical exemple of the advection test computed the scheme (\ref{['eq:b8']}) at time $t_0=0\Delta t$, $t_1=30\Delta t$, $t_2=45\Delta t$, $t_3=60\Delta t$, $t_4=75\Delta t$ then $t_5=90\Delta t$ ($N=64$ and $\Delta t=0.1$). Before time $t\approx t_2$, the solution is correct. The sign of the electric field is reversed at $t=4.5$. Then a numerical instability starts to be visible for $t\approx t_2$, diminishes at later times, but is fully unacceptable at the final time.
• Figure 2: Results of the advection test computed the scheme (\ref{['eq:b8']}) at time $t_0=0\Delta t$ to $t_5=90\Delta t$ ($N=64$ and $\Delta t=0.1$) with the first method.
• Figure 3: Results of the advection test computed the scheme (\ref{['eq:b8']}) at time $t_0=0\Delta t$ to $t_5=90\Delta t$ ($N=64$ and $\Delta t=0.1$) with the second method (with $\varepsilon=10^{-10}$).
• Figure 4: 2D diocotron instability: Evolution of electron charge density over time utilizing three distinct cases: the original method without stabilization (first row), the first method (second row), and the first method that implements the simplification explained in Remark \ref{['rem:remi']}, which is a numerically conservative method (third row). These visualisations are based on a $64^2$ mesh grid. Each figure's color bar displays the respective minimum and maximum values of the charge densities.
• Figure 5: Density function at time $t=0$ and $t=20$. On the left the original method. On the right the stabilization with the first method.

Theorems & Definitions (44)

• Remark 2.1
• Remark 2.2
• Definition 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Remark 3.4
• Lemma 3.5
• proof
• Theorem 4.1