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首页> 外文期刊>Journal of Computational Physics >Overcoming numerical shockwave anomalies using energy balanced numerical schemes. Application to the Shallow Water Equations with discontinuous topography
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Overcoming numerical shockwave anomalies using energy balanced numerical schemes. Application to the Shallow Water Equations with discontinuous topography

机译：使用能量平衡数值方案克服数值冲击波异常。 用不连续地形应用于浅水方程

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摘要

When designing a numerical scheme for the resolution of conservation laws, the selection of a particular source term discretization (STD) may seem irrelevant whenever it ensures convergence with mesh refinement, but it has a decisive impact on the solution. In the framework of the Shallow Water Equations (SWE), well-balanced STD based on quiescent equilibrium are unable to converge to physically based solutions, which can be constructed considering energy arguments. Energy based discretizations can be designed assuming dissipation or conservation, but in any case, the STD procedure required should not be merely based on ad hoc approximations. The STD proposed in this work is derived from the Generalized Hugoniot Locus obtained from the Generalized Rankine Hugoniot conditions and the Integral Curve across the contact wave associated to the bed step. In any case, the STD must allow energy-dissipative solutions: steady and unsteady hydraulic jumps, for which some numerical anomalies have been documented in the literature. These anomalies are the incorrect positioning of steady jumps and the presence of a spurious spike of discharge inside the cell containing the jump. The former issue can be addressed by proposing a modification of the energy-conservative STD that ensures a correct dissipation rate across the hydraulic jump, whereas the latter is of greater complexity and cannot be fixed by simply choosing a suitable STD, as there are more variables involved. The problem concerning the spike of discharge is a well-known problem in the scientific community, also known as slowly-moving shock anomaly, it is produced by a nonlinearityof the Hugoniot locus connecting the states at both sides of the jump. However, it seems that this issue is more a feature than a problem when considering steady solutions of the SWE containing hydraulic jumps. The presence of the spurious spike in the discharge has been taken for granted and has become a feature of the solution. Even though it does not disturb the rest of the solution in steady cases, when considering transient cases it produces a very undesirable shedding of spurious oscillations downstream that should be circumvented. Based on spike-reducing techniques (originally designed for homogeneous Euler equations) that propose the construction of interpolated fluxes in the untrustworthy regions, we design a novel Roe-type scheme for the SWE with discontinuous topography that reduces the presence of the aforementioned spurious spike. The resulting spike-reducing method in combination with the proposed STD ensures an accurate positioning of steady jumps, provides convergence with mesh refinement, which was not possible for previous methods that cannot avoid the spike. (C) 2017 Elsevier Inc. All rights reserved.
机译：在为解决保护法的分辨率设计数值方案时，每当物质细化时，可以选择特定源期限离散化（STD）的选择可能看起来无关，但它对解决方案具有决定性的影响。在浅水方程式（SWE）的框架中，基于静态平衡的良好平衡的STD无法收敛到物理基础的解决方案，这可以考虑能量参数来构造。可以假设耗散或保护可以设计能量的离散化，但在任何情况下，所需的STD程序不应仅仅基于临时近似。在这项工作中提出的STD源自从广义朗肯队的普通条件和与床台相关的接触波的整体曲线获得的广义Hugoniot轨迹源自。无论如何，STD必须允许节能解决方案：稳定和不稳定的液压跳跃，其中一些数值异常在文献中被记录在内。这些异常是稳定跳跃的不正确定位以及含有跳跃的电池内的放电的虚假尖峰。可以通过提出能量保守的STD的修改来解决前一个问题，以确保液压跳跃的正确耗散速率，而后者具有更大的复杂性，并且不能通过简单地选择合适的STD来固定，因为有更多的变量涉及。关于放电尖峰的问题是科学界的一个众所周知的问题，也称为缓慢移动的休克异常，它是由Hugoniot轨迹的非线性的非线性产生的，将状态连接在跳跃的两侧。然而，在考虑含有液压跳跃的SWE的稳定解决方案时，似乎这个问题比问题更重要。在放电中存在杂散尖峰的存在已被认为是理所当然的并且已成为解决方案的特征。即使在稳定的情况下，在稳定情况下没有打扰其余的解决方案，当考虑到瞬态案例，它产生了应该被避难的虚假振荡的非常不希望的脱落。基于峰值减少技术（最初为均匀的欧拉方程设计）提出了不值得信赖的区域中的内插通量的构建，我们设计了一种具有不连续地形的SWE的新型Roe型方案，这减少了上述杂散尖峰的存在。由此产生的尖峰减少方法与所提出的STD结合确保了稳定跳跃的精确定位，提供了与网眼细化的收敛，这对于之前无法避免尖峰的方法是不可能的。 （c）2017年Elsevier Inc.保留所有权利。

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