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.

翻译：我们分析了使用非对称埃尔米特基函数对线性输运方程进行离散化时，为何可能在二次范数意义下不稳定。其主要原因在于，无限矩线性系统的有限截断会丢失关于格拉姆矩阵的斜对称性质。随后，我们提出了一个关于任意一对非对称基函数标量积的原创闭式表达式。该表达式使得能够对线性系统进行两种简单修正，从而恢复斜对称性。通过构造，新方法在自然$L^2$范数下具有二次稳定性。我们阐述了如何将方法推广至数值等离子体物理中遇到的其他输运方程。针对不同性质的振荡电场开展的基础数值试验，验证了我们算法的无条件稳定性特性。