黎曼问题中波速的界：直接理论估计
Bounds for Wave Speeds in the Riemann Problem: Direct Theoretical Estimates
摘要：
在本文中，我们提供了两个最快的波速出现的解决方案的黎曼问题为三个著名的双曲型系统，即欧拉方程的气体动力学，浅水方程和动脉血流方程的边界估计。提出了几种方法，都是直接的，这是非迭代。由此产生的界限范围从粗糙但简单的估计，准确但复杂的估计，使有限的使用信息的解决方案的黎曼问题。通过一套精心挑选的测试问题，我们评估我们的波速估计对精确解和以前提出的波速估计。结果证实，推导出的理论界限实际上是如此，从下面和上面，分别为最小和最大的波速。结果还表明，流行的先前提出的估计不约束的真实波速一般。考虑的应用，但不追求在这里，包括（i）可靠的实现的柯朗条件，以确定一个稳定的时间步长在所有显式方法的双曲型方程;（ii）使用本地时间步进算法和（iii）建设的HLL型数值通量的双曲型方程。
英文摘要：
In this paper we provide bound estimates for the two fastest wave speeds emerging from the solution of the Riemann problem for three well-known hyperbolic systems, namely the Euler equations of gas dynamics, the shallow water equations and the blood flow equations for arteries. Several approaches are presented, all being direct, that is non-iterative. The resulting bounds range from crude but simple estimates to accurate but sophisticated estimates that make limited use of information from the solution of the Riemann problem. Through a carefully chosen suite of test problems we asses our wave speed estimates against exact solutions and against previously proposed wave speed estimates. The results confirm that the derived theoretical bounds are actually so, from below and above, for minimal and maximal wave speeds respectively. The results also show that popular previously proposed estimates do not bound the true wave speeds in general. Applications in mind, but not pursued here, include (i) reliable implementation of the Courant condition to determine a stable time step in all explicit methods for hyperbolic equations; (ii) use in local time stepping algorithms and (iii) construction of HLL-type numerical fluxes for hyperbolic equations.
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