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Why many theories of shock waves are necessary: Kinetic functions, equivalent equations, and fourth-order models

机译：为什么需要许多冲击波理论：动力学函数，等效方程式和四阶模型

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摘要

We consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena. These models admit under-compressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation. Building on earlier work by LeFloch and collaborators, we investigate the numerical approximation of these models by high-order finite difference schemes, and uncover several new features of the kinetic function associated with physically motivated second and third-order regularization terms, especially viscosity and capillarity terms. On one hand, the role of the equivalent equation associated with a finite difference scheme is discussed. We conjecture here and demonstrate numerically that the (numerical) kinetic function associated with a scheme approaches the (analytic) kinetic function associated with the given model - especially since its equivalent equation approaches the regularized model at a higher order. On the other hand, we demonstrate numerically that a kinetic function can be associated with the thin liquid film model and the generalized Camassa-Hohn model. Finally, we investigate to what extent a kinetic function can be associated with the equations of van der Waals fluids, whose flux-function admits two inflection points. (C) 2008 Elsevier Inc. All rights reserved.
机译：我们考虑了几种非线性双曲守恒律系统，它们描述了存在相变现象时非线性波的动力学。这些模型允许不由标准熵准则唯一确定的欠压缩冲击波，而必须通过动力学关系来表征。在LeFloch和合作者的早期工作的基础上，我们通过高阶有限差分方案研究了这些模型的数值逼近，并发现了与物理激励的第二阶和第三阶正则化项相关的动力学函数的几个新特征，尤其是粘度和毛细管条款。一方面，讨论了与有限差分方案相关的等价方程的作用。我们在这里进行猜想并在数值上证明与方案关联的（数值）动力学函数接近与给定模型关联的（解析）动力学函数-尤其是因为其等效方程以更高阶接近正则化模型。另一方面，我们通过数值证明动力学函数可以与薄膜模型和广义Camassa-Hohn模型相关联。最后，我们研究动力学函数可以在多大程度上与范德华流体方程相关联，范德华流体的通量函数允许两个拐点。 （C）2008 Elsevier Inc.保留所有权利。

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