A Finite Volume Implicit Euler Scheme for the Linearized Shallow Water Equations: Stability and Convergence

The linearized shallow water equations are discretized in space by a finite volume method and in time by an implicit Euler scheme. Stability and convergence of the scheme are proved.     

An improved algorithm for the shallow water equations model reduction: Dynamic Mode Decomposition vs POD

We propose an improved framework for dynamic mode decomposition (DMD) of 2‐D flows for problems originating from meteorology when a large time step acts like a filter in obtaining the significant Koopman modes, therefore, the classic DMD method is not effective. This study is motivated by the need to further clarify the connection between Koopman modes and proper orthogonal decomposition (POD) dynamic modes. We apply DMD and POD to derive reduced order models (ROM) of the shallow water equations. Key innovations for the DMD‐based ROM introduced in this paper are the use of the Moore–Penrose pseudoinverse in the DMD computation that produced an accurate result and a novel selection method for the DMD modes and associated amplitudes and Ritz values. A quantitative comparison of the spatial modes computed from the two decompositions is performed, and a rigorous error analysis for the ROM models obtained is presented. Copyright © 2015 John Wiley & Sons, Ltd.     

梯级溃坝洪水洪峰增强机制

我国在多条河流上修建了大量梯级水库, 梯级坝溃决诱发洪水大大超过单坝溃决洪水洪峰, 因此亟需加深对梯级坝溃决洪水洪峰增强机制的认识. 本文建立了梯级坝溃决洪水演进过程的一维浅水动力学模型, 发展了一套能捕捉激波、干湿边界和保平衡结构的数值求解方法, 通过大量算例, 系统研究了梯级坝溃决洪水演进过程的质量转化和能量转化机制. 研究结果表明, 梯级溃决中, 上游溃决诱发的洪水大大增大下游水库的质量和动量, 形成一个带动量的水塔, 同时在尾部残留一个动量较大的射流, 不断补充下游坝体溃决后水塔的质量和动量, 持续维持洪峰高度. 根据该射流-水塔机制, 建立了梯级坝溃决洪水演进过程对应的射流-水塔单坝溃决洪水过程等效模型, 该等效模型基本反映了梯级坝溃决诱发洪水的洪峰过程, 并成功预测了多个坝间距为百公里量级的梯级坝溃决洪水洪峰高程和流量, 可望为流域防洪和梯级坝设计提供理论依据.      

A well‐balanced explicit/semi‐implicit finite element scheme for shallow water equations in drying–wetting areas

Numerical solutions of the shallow water equations can be used to reproduce flow hydrodynamics occurring in a wide range of regions. In hydraulic engineering, the objectives include the prediction of dam break wave propagation, fluvial floods and other catastrophic flooding phenomena, the modeling of estuarine and coastal circulations, and the design and optimization of hydraulic structures. In this paper, a well‐balanced explicit and semi‐implicit finite element scheme for shallow water equations over complex domains involving wetting and drying is proposed. The governing equations are discretized by a fractional finite element method using a two‐step Taylor–Galerkin scheme. First, the intermediate increment of conserved variable is obtained explicitly neglecting the pressure gradient term. This is then corrected for the effects of pressure once the pressure increment has been obtained from the Poisson equation. In order to maintain the ‘well‐balanced’ property, the pressure gradient term and bed slope terms are incorporated into the Poisson equation. Moreover, a local bed slope modification technique is employed in drying–wetting interface treatments. The proposed model is well validated against several theoretical benchmark tests. Copyright © 2014 John Wiley & Sons, Ltd.     

A mass conservative well‐balanced reconstruction at wet/dry interfaces for the Godunov‐type shallow water model

This paper presents a novel mass conservative, positivity preserving wetting and drying treatment for Godunov‐type shallow water models with second‐order bed elevation discretization. The novel method allows to compute water depths equal to machine accuracy without any restrictions on the time step or any threshold that defines whether the finite volume cell is considered to be wet or dry. The resulting scheme is second‐order accurate in space and keeps the C‐property condition at fully flooded area and also at the wet/dry interface. For the time integration, a second‐order accurate Runge–Kutta method is used. The method is tested in two well‐known computational benchmarks for which an analytical solution can be derived, a C‐property benchmark and in an additional example where the experimental results are reproduced. Overall, the presented scheme shows very good agreement with the reference solutions. The method can also be used in the discontinuous Galerkin method. Copyright © 2016 John Wiley & Sons, Ltd.     

A mass‐conservative staggered immersed boundary model for solving the shallow water equations on complex geometries

In this work, an approach is proposed for solving the 3D shallow water equations with embedded boundaries that are not aligned with the underlying horizontal Cartesian grid. A hybrid cut‐cell/ghost‐cell method is used together with a direction‐splitting implicit solver: Ghost cells are used for the momentum equations in order to prescribe the correct boundary condition at the immersed boundary, while cut cells are used in the continuity equation in order to conserve mass. The resulting scheme is robust, does not suffer any time step limitation for small cut cells, and conserves fluid mass up to machine precision. Moreover, the solver displays a second‐order spatial accuracy, both globally and locally. Comparisons with analytical solutions and reference numerical solutions on curvilinear grids confirm the quality of the method. Copyright © 2015 John Wiley & Sons, Ltd.     

A dual‐weighted trust‐region adaptive POD 4‐D Var applied to a finite‐volume shallow water equations model on the sphere

In this paper we study solutions of an inverse problem for a global shallow water model controlling its initial conditions specified from the 40‐yr ECMWF Re‐analysis (ERA‐40) data sets, in the presence of full or incomplete observations being assimilated in a time interval (window of assimilation) with or without background error covariance terms. As an extension of the work by Chen et al. (Int. J. Numer. Meth. Fluids 2009), we attempt to obtain a reduced order model of the above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4D‐Var for a finite volume global shallow water equation model based on the Lin–Rood flux‐form semi‐Lagrangian semi‐implicit time integration scheme. Different approaches of POD implementation for the reduced inverse problem are compared, including a dual‐weighted method for snapshot selection coupled with a trust‐region POD adaptivity approach. Numerical results with various observational densities and background error covariance operator are also presented. The POD 4‐D Var model results combined with the trust‐region adaptivity exhibit similarity in terms of various error metrics to the full 4D Var results, but are obtained using a significantly lesser number of minimization iterations and require lesser CPU time. Based on our previous and current work, we conclude that POD 4‐D Var certainly warrants further studies, with promising potential of its extension to operational 3‐D numerical weather prediction models. Copyright © 2011 John Wiley & Sons, Ltd.     

An efficient splitting technique for two‐layer shallow‐water model

We consider the numerical approximation of the weak solutions of the two‐layer shallow‐water equations. The model under consideration is made of two usual one‐layer shallow‐water model coupled by nonconservative products. Because of the nonconservative products of the system, which couple both one‐layer shallow‐water subsystems, the usual numerical methods have to consider the full model. Of course, uncoupled numerical techniques, just involving finite volume schemes for the basic shallow‐water equations, are very attractive since they are very easy to implement and they are costless. Recently, a stable layer splitting technique was introduced [Bouchut and Morales de Luna, M2AN Math Model Numer Anal 42 (2008), 683–698]. In the same spirit, we exhibit new splitting technique, which is proved to be well balanced and non‐negative preserving. The main benefit issuing from the here derived uncoupled method is the ability to correctly approximate the solution of very severe benchmarks. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1396–1423, 2015     

浅埋隧道围岩受施工扰动的时空变化规律研究

采用监测分析和数值模拟相结合的方法, 详细模拟了浅埋隧道施工的全过程, 得到不同施工情况下围岩的应力应变状态, 并深入分析围岩受施工扰动的应力变化规律和沉降变形趋势, 在此基础上, 采用非线性回归方法并考虑时间影响因素, 得到了地表沉降的时空变形规律.