计算精确稳定性条件
Computing the refined stability condition
摘要：
激波的经典（无粘）稳定性分析是基于洛帕廷斯基行列式，Δ-频率的函数，其零点决定了潜在激波的稳定性。对Δ的仔细分析表明，在某些情况下，参数空间的稳定区域和不稳定区域被参数的开集分隔开。Zumbrun和Serre [印第安纳大学数学杂志，48（1999）937- 992]已经表明，通过考虑在Δ的定义中不存在的粘性效应，有可能确定在参数空间的开放中性集合中失去稳定性的精确位置。特别是，他们表明，适当的局部扰动下的过渡到不稳定性是由“有效粘度”系数。在这里，在最简单的可能的设置，我们提出并实现了两种新的方法对这个系数的实际计算。此外，在一种特殊情况下，我们还得到了相应的微分方程的精确解。
英文摘要：
The classical (inviscid) stability analysis of shock waves is based on the Lopatinski determinant, Δ---a function of frequencies whose zeros determine the stability of the underlying shock. A careful analysis of Δ shows that in some cases the stable and unstable regions of parameter space are separated by an open set of parameters. Zumbrun and Serre [Indiana Univ. Math. J., 48 (1999) 937--992] have shown that, by taking account of viscous effects not present in the definition of Δ, it is possible to determine the precise location in the open, neutral set of parameter space at which stability is lost. In particular, they show that the transition to instability under suitably localized perturbations is determined by an "effective viscosity" coefficient. Here, in the simplest possible setting, we propose and implement two new approaches toward the practical computation of this coefficient. Moreover, in a special case, we derive an exact solution of the relevant differential equations.
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