凸FPU链的黎曼解算子和欠压缩激波
Riemann solvers and undercompressive shocks of convex FPU chains
摘要：
我们考虑具有一般凸势的FPU型原子链。双曲空间-时间标度中的朴素连续极限是质量和动量守恒的p-系统。我们系统地比较黎曼解的p-系统与数值解的离散黎曼问题的FPU链，并认为后者可以描述修改的p-系统黎曼解算子。我们允许通量有一个转折点，并观察第三种类型的基本波（保守冲击）的原子模拟。这些波是异宿行波，对应于p系统的非经典欠压缩冲击。我们分析这样的冲击通量与一个或多个转折点。根据通量的凸性，我们提出FPU-黎曼解算子。我们的数值模拟证实了Lax激波被所谓的色散激波所取代。对于凹凸通量，我们提供了数值证据，凸FPU链遵循p-系统产生保守的超音速激波。然而，对于凹凸通量，保守的冲击的p-系统是亚音速的，并没有出现在FPU-Riemann解。
英文摘要：
We consider FPU-type atomic chains with general convex potentials. The naive continuum limit in the hyperbolic space-time scaling is the p-system of mass and momentum conservation. We systematically compare Riemann solutions to the p-system with numerical solutions to discrete Riemann problems in FPU chains, and argue that the latter can be described by modified p-system Riemann solvers. We allow the flux to have a turning point, and observe a third type of elementary wave (conservative shocks) in the atomistic simulations. These waves are heteroclinic travelling waves and correspond to non-classical, undercompressive shocks of the p-system. We analyse such shocks for fluxes with one or more turning points. Depending on the convexity properties of the flux we propose FPU-Riemann solvers. Our numerical simulations confirm that Lax-shocks are replaced by so called dispersive shocks. For convex-concave flux we provide numerical evidence that convex FPU chains follow the p-system in generating conservative shocks that are supersonic. For concave-convex flux, however, the conservative shocks of the p-system are subsonic and do not appear in FPU-Riemann solutions.
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