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首页> 外文期刊>Journal of Computational Physics >A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers
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A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

机译：基于松弛的单相和两相浅流模型的黎曼求解器。与Roe和VFRoe求解器的关系

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摘要

We present a Riemann solver derived by a relaxation technique for classical single-phase shallow flow equations and for a two-phase shallow flow model describing a mixture of solid granular material and fluid. Our primary interest is the numerical approximation of this two-phase solid/fluid model, whose complexity poses numerical difficulties that cannot be efficiently addressed by existing solvers. In particular, we are concerned with ensuring a robust treatment of dry bed states. The relaxation system used by the proposed solver is formulated by introducing auxiliary variables that replace the momenta in the spatial gradients of the original model systems. The resulting relaxation solver is related to Roe solver in that its Riemann solution for the flow height and relaxation variables is formally computed as Roe's Riemann solution. The relaxation solver has the advantage of a certain degree of freedom in the specification of the wave structure through the choice of the relaxation parameters. This flexibility can be exploited to handle robustly vacuum states, which is a well known difficulty of standard Roe's method, while maintaining Roe's low diffusivity. For the single-phase model positivity of flow height is rigorously preserved. For the two-phase model positivity of volume fractions in general is not ensured, and a suitable restriction on the CFL number might be needed. Nonetheless, numerical experiments suggest that the proposed two-phase flow solver efficiently models wet/dry fronts and vacuum formation for a large range of flow conditions.As a corollary of our study, we show that for single-phase shallow flow equations the relaxation solver is formally equivalent to the VFRoe solver with conservative variables of Gallou?t and Masella [T. Gallou?t, J.-M. Masella, Un schéma de Godunov approché C.R. Acad. Sci. Paris, Série I, 323 (1996) 77-84]. The relaxation interpretation allows establishing positivity conditions for this VFRoe method.
机译：我们介绍了通过松弛技术派生的Riemann求解器，用于经典单相浅流方程和描述固体颗粒材料和流体混合物的两相浅流模型。我们的主要兴趣是此两相固/流体模型的数值逼近，其复杂度带来数值难题，而现有求解器无法有效解决这些难题。特别地，我们关注确保对干床状态的有效处理。拟议求解器使用的松弛系统是通过引入辅助变量来代替原始模型系统的空间梯度中的矩而制定的。所得的松弛求解器与Roe求解器有关，因为它的流动高度和松弛变量的Riemann解正式计算为Roe的Riemann解。通过选择弛豫参数，弛豫求解器在波动结构的规范方面具有一定的自由度。可以利用这种灵活性来处理鲁棒的真空状态，这是标准Roe方法的众所周知的困难，同时又要保持Roe的低扩散率。对于单相模型，严格保留流动高度的正值。对于两相模型，通常不能确保体积分数的正性，可能需要对CFL数进行适当的限制。但是，数值实验表明，所提出的两相流求解器可以有效地对大范围流动条件下的湿/干前沿和真空形成进行建模。作为我们研究的推论，我们表明对于单相浅流方程，松弛求解器在形式上等效于具有Gallou？t和Masella [T.的保守变量的VFRoe求解器。加洛特，J.-M。 Masella，Goschömade Godunov副总裁C.R. Acad。科学巴黎，塞丽（SérieI），323（1996）77-84]。弛豫解释允许为此VFRoe方法建立阳性条件。

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