Shock Wave Admissibility for Quadratic Conservation Laws
Shock Wave Admissibility for Quadratic Conservation Laws
复制标题
二次守恒定律的冲击波容许性
DOI：
10.1006/jdeq.1995.1075
复制
发表时间：
1995
影响因子：
2.4
通讯作者：
B. Plohr
中科院分区：
文献类型：
--
作者：
S. Čanić;B. Plohr
摘要本文提出了一种新的方法来研究变类型守恒律系统的容许激波解的稳定性。我们处理的系统具有二次通量函数。我们采用的基本波流形W作为一个全球性的框架来表征冲击波，符合粘性容许性准则。W的点使与冲击波解相关的动力系统参数化。W中包含可允许冲击波的区域以结构不稳定动力系统的轨迹为界。显式公式与鞍结，霍普夫，Bogdanov-Takens分歧，并与直线异宿连接的轨迹。利用Melnikov积分分析，计算了在W .此外，利用数值方法，我们还研究了曲线连接轨对应的异宿轨线和完全同宿轨线。我们发现一个通用的，二维切片的基波流形的区域的容许波，并将其与一组符合拉克斯容许标准的激波点，从而阐明如何不同的粘性剖面容许这个标准。
Abstract In this work we present a new approach to the study of the stability of admissible shock wave solutions for systems of conservation laws that change type. The systems we treat have quadratic flux functions. We employ the fundamental wave manifold W as a global framework to characterize shock waves that comply with the viscosity admissibility criterion. Points of W parametrize dynamical systems associated with shock wave solutions. The region of W comprising admissible shock waves is bounded by the loci of structurally unstable dynamical systems. Explicit formulae are presented for the loci associated with saddle-node, Hopf, and Bogdanov-Takens bifurcation, and with straight-line heteroclinic connections. Using Melnikov′s integral analysis, we calculate the tangent to the homoclinic part of the admissibility boundary at Bogdanov-Takens points of W . Furthermore, using numerical methods, we explore the heteroclinic loci corresponding to curved connecting orbits and the complete homoclinic locus. We find the region of admissible waves for a generic, two-dimensional slice of the fundamental wave manifold, and compare it with the set of shock points that comply with the Lax admissibility criterion, thereby elucidating how this criterion differs from viscous profile admissibility.