标题： Rusanov型格式用于双曲方程：波速估计、单调性和稳定性 显示英文标题

标题： Rusanov-type Schemes for Hyperbolic Equations: Wave-Speed Estimates, Monotonicity and Stability

摘要： HLL型格式构成了一个大型的数值方法层次结构，在有限体积和不连续伽辽金有限元框架中用于求解双曲方程。 该通量层次结构包括Rusanov格式、HLL格式、HLLC格式和其他变体。 所有这些格式都依赖于波速估计。 最近的研究表明，目前使用的大多数波速估计都低估了真实的波速。 在本文中，我们对波速估计中的误差对所推导格式的单调性和稳定性特性产生的后果进行了理论研究。 对于层次结构中最简单的情况，即Rusanov型格式，我们根据一维和二维空间中的线性对流方程进行了详细分析。 发现波速低估的误差可能导致单调性的丧失、稳定极限的严重降低，甚至稳定性的丧失。 高估的误差虽然保持了单调性，但会导致稳定极限的降低。 我们发现，高估比低估更好，有两个原因。 首先，高估的格式是单调的，其次，对于与精确速度等量偏离的情况，它们的稳定区域比低估的大。 本文的发现可能有助于提高对看似简单的实际计算任务——提供波速估计——潜在陷阱的认识。 我们报告的发现也可能激发后续对复杂非线性双曲系统的进一步研究，这些系统需要对两个或多个波进行估计，例如在HLL和HLLC格式中。 显示英文摘要

摘要： HLL-type schemes constitute a large hierarchy of numerical methods, in the finite volume and discontinuous Galerkin finite element frameworks, for solving hyperbolic equations. The hierarchy of fluxes includes Rusanov schemes, HLL schemes, HLLC schemes, and other variations. All of these schemes rely on wave speed estimates. Recent work has shown that most wave speed estimates in current use underestimate the true wave speeds. In the present paper we carry out a theoretical study of the consequences arising from errors in the wave speed estimates, on the monotonicity and stability properties of the derived schemes. For the simplest case of the hierarchy, that is Rusanov-type schemes, we carry out a detailed analysis in terms of the linear advection equation in one and two space dimensions. It is found that errors from underestimates of the wave speed could cause loss of monotonicity, severe reduction of the stability limit, and even loss of stability. Errors from overestimates, though preserving monotonicity, will cause a reduction of the stability limit. We find that overestimation is preferable to underestimation, for two reasons. First, schemes from overestimation are monotone, and second, their stability regions are larger than those from underestimation, for equivalent displacements from the exact speed. The findings of this paper may prove useful in raising awareness of the potential pitfalls of a seemingly simple practical computational task, that of providing wave speed estimates. Our reported findings may also motivate subsequent studies for complex non-linear hyperbolic systems, requiring estimates for two or more waves, such as in HLL and HLLC schemes.