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 illustrate the unconditional stability properties of our algorithms.

翻译：我们分析了为什么使用非对称埃尔米特基函数离散线性输运问题在二次范数下可能不稳定的原因。主要原因是无限矩线性系统的有限截断丢失了相对于格拉姆矩阵的斜对称性。随后，我们提出了一种封闭公式，用于计算任意一对非对称基函数的标量积。该公式使得构建两种简单的线性系统修改成为可能，从而恢复斜对称性。通过构造，新方法在自然$L^2$范数下是二次稳定的。我们解释了如何将这一方法推广到数值等离子体物理中遇到的其他输运方程。基本数值测试展示了我们算法的无条件稳定性特性。