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

TL;DR

This paper investigates the instability caused by asymmetric Hermite basis discretization in linear transport problems and introduces modifications that restore skew-symmetry, ensuring quadratic stability in plasma physics applications.

Contribution

It provides a novel closed-form formula for scalar products of asymmetric basis functions and develops stable linear system modifications for transport equations.

Findings

Modified systems recover skew-symmetry

Algorithms are quadratically stable in L^2 norm

Numerical tests confirm unconditional stability

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.