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.     

Low-order modelling of shallow water equations for sensitivity analysis using proper orthogonal decomposition

This article presents a reduced-order model (ROM) of the shallow water equations (SWEs) for use in sensitivity analyses and Monte-Carlo type applications. Since, in the real world, some of the physical parameters and initial conditions embedded in free-surface flow problems are difficult to calibrate accurately in practice, the results from numerical hydraulic models are almost always corrupted with uncertainties. The main objective of this work is to derive a ROM that ensures appreciable accuracy and a considerable acceleration in the calculations so that it can be used as a surrogate model for stochastic and sensitivity analyses in real free-surface flow problems. The ROM is derived using the proper orthogonal decomposition (POD) method coupled with Galerkin projections of the SWEs, which are discretised through a finite-volume method. The main difficulty of deriving an efficient ROM is the treatment of the nonlinearities involved in SWEs. Suitable approximations that provide rapid online computations of the nonlinear terms are proposed. The proposed ROM is applied to the simulation of hypothetical flood flows in the Bordeaux breakwater, a portion of the ‘Rivière des Prairies' located near Laval (a suburb of Montreal, Quebec). A series of sensitivity analyses are performed by varying the Manning roughness coefficient and the inflow discharge. The results are satisfactorily compared to those obtained by the full-order finite volume model.     

A POD-based reduced-order finite difference extrapolating model with fully second-order accuracy for non-stationary Stokes equations

A proper orthogonal decomposition (POD) reduced-order finite difference (FD) extrapolating model is established for the channel flow with local expansion denoted by non-stationary Stokes equations. The POD-based reduced-order numerical model to produce the solutions on the time span [T₀, T] (T₀ ? T) are obtained by extrapolation and iteration from the very short time span [0, T₀] information. The guides to choose the number of POD basis and renew POD basis are provided, and an implementation for solving the POD-based reduced-order FD extrapolating model is given. Some numerical experiments are used to show that the POD-based reduced-order FD extrapolating model is feasible and efficient for simulating the channel flow with local expansion.     

A modified lattice Boltzmann model for shallow water flows over complex topography

A modified lattice Boltzmann model is proposed to describe shallow water flows over complex topography. In the proposed model, the quadratic depth term is excluded from the equilibrium distribution functions (EDFs), and the hydrostatic pressure term is combined with the bed slope term to be treated as a part of the sourcing term in the lattice Boltzmann equation (LBE). Therefore, it is unnecessary to match the coefficients of the quadratic depth term in the EDFs with those of the bed slope term in the sourcing terms in the LBE. This would bring more flexibility to the treatment of the sourcing terms in the LBE. In order to recover the shallow water equations (SWEs), the basic constraints are redefined, and under these constraints, the coefficients of the EDFs are derived afterwards. Several benchmark problems are used to validate the proposed model, including stationary case, steady flows over a two‐dimensional bump and tidal wave flows over irregular bed elevation. The computed results are in excellent agreement with the results of the other numerical methods and the analytical solutions, indicating that the proposed model is capable of simulating shallow water flows over complex bathymetry. It also proves that the proposed model has potential to produce competitive solutions to shallow water flows over complex bed topography. Copyright © 2014 John Wiley & Sons, Ltd.     

Interpolation-based reduced-order modelling for steady transonic flows via manifold learning

This paper presents a parametric reduced-order model (ROM) based on manifold learning (ML) for use in steady transonic aerodynamic applications. The main objective of this work is to derive an efficient ROM that exploits the low-dimensional nonlinear solution manifold to ensure an improved treatment of the nonlinearities involved in varying the inflow conditions to obtain an accurate prediction of shocks. The reduced-order representation of the data is derived using the Isomap ML method, which is applied to a set of sampled computational fluid dynamics (CFD) data. In order to develop a ROM that has the ability to predict approximate CFD solutions at untried parameter combinations, Isomap is coupled with an interpolation method to capture the variations in parameters like the angle of attack or the Mach number. Furthermore, an approximate local inverse mapping from the reduced-order representation to the full CFD solution space is introduced. The proposed ROM, called Isomap+I, is applied to the two-dimensional NACA 64A010 airfoil and to the 3D LANN wing. The results are compared to those obtained by proper orthogonal decomposition plus interpolation (POD+I) and to the full-order CFD model.     

A stabilized SPH method for inviscid shallow water flows

In this paper, the smoothed particle hydrodynamics (SPH) method is applied to the solution of shallow water equations. A brief review of the method in its standard form is first described then a variational formulation using SPH interpolation is discussed. A new technique based on the Riemann solver is introduced to improve the stability of the method. This technique leads to better results. The treatment of solid boundary conditions is discussed but remains an open problem for general geometries. The dam‐break problem with a flat bed is used as a benchmark test. Copyright © 2004 John Wiley & Sons, Ltd.     

Principal interval decomposition framework for POD reduced‐order modeling of convective Boussinesq flows

A principal interval decomposition (PID) approach is presented for the reduced‐order modeling of unsteady Boussinesq equations. The PID method optimizes the lengths of the time windows over which proper orthogonal decomposition (POD) is performed and can be highly effective in building reduced‐order models for convective problems. The performance of these POD models with and without using the PID approach is investigated by applying these methods to the unsteady lock‐exchange flow problem. This benchmark problem exhibits a strong shear flow induced by a temperature jump and results in the Kelvin–Helmholtz instability. This problem is considered a challenging benchmark problem for the development of reduced‐order models. The reference solutions are obtained by direct numerical simulations of the vorticity and temperature transport equations using a compact fourth‐order‐accurate scheme. We compare the accuracy of reduced‐order models developed with different numbers of POD basis functions and different numbers of principal intervals. A linear interpolation model is constructed to obtain basis functions when varying physical parameters. The predictive performance of our models is then analyzed over a wide range of Reynolds numbers. It is shown that the PID approach provides a significant improvement in accuracy over the standard Galerkin POD reduced‐order model. This numerical assessment of the PID shows that it may represent a reliable model reduction tool for convection‐dominated, unsteady‐flow problems. 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.     

Boundary discretization for high‐order discontinuous Galerkin computations of tidal flows around shallow water islands

In this paper some preliminary results concerning the application of the high‐order discontinuous Galerkin (DG) method for the resolution of realistic problems of tidal flows around shallow water islands are presented. In particular, tidal flows are computed around the Rattray island located in the Great Barrier Reef. This island is a standard benchmark problem well documented in the literature providing useful in situ measurements for validation of the model. Realistic elements of the simulation are a tidal flow forcing, a variable bathymetry and a non‐trivial coastline. The computation of tidal flows in shallow water around an island is very similar to the simulation of the Euler equations around bluff bodies in quasi‐steady flows. The main difference lies in the high irregularity of islands' shapes and in the fact that, in the framework of large‐scale ocean models, the number of elements to represent an island is drastically limited compared with classical engineering computations. We observe that the high‐order DG method applied to shallow water flows around bluff bodies with poor linear boundary representations produces oscillations and spurious eddies. Surprisingly those eddies may have the right size and intensity but may be generated by numerical diffusion and are not always mathematically relevant. Although not interested in solving accurately the boundary layers of an island, we show that a high‐order boundary representation is mandatory to avoid non‐physical eddies and spurious oscillations. It is then possible to parametrize accurately the subgrid‐scale processes to introduce the correct amount of diffusion in the model. The DG results around the Rattray island are eventually compared with current measurements and reveal good agreement. Copyright © 2008 John Wiley & Sons, Ltd.     

Lattice Boltzmann modeling of shallow water flows over discontinuous beds

Numerical modeling of shallow water flows over discontinuous beds is presented. The flows are described with the shallow water equations and the equations are solved using the lattice Boltzmann method (LBM) with single relaxation time (Bhatnagar–Gross–Krook‐LBM (BGK‐LBM)) and the multiple relaxation time (MRT‐LBM). The weighted centered scheme for force term together with the bed height for a bed slope is described to improve simulation of flows over discontinuous bed. Furthermore, the resistance stress is added to include the local head loss caused by flow over a step. Four test cases, one‐dimensional tidal over regular bed and steps, dam‐break flows, and two‐dimensional shallow water flow over a square block, are considered to verify the present method. Agreements between predictions and analytical solutions are satisfactory. Furthermore, the performance and CPU cost time of BGK‐LBM and MRT‐LBM are compared and studied. The results have shown that the lattice Boltzmann method is simple and accurate for simulating shallow water flows over discontinuous beds. This demonstrates the capability and applicability of the lattice Boltzmann method in modeling shallow water flows on bed topography with a discontinuity in practical hydraulic engineering. Copyright © 2014 John Wiley & Sons, Ltd.     

High-resolution numerical model for shallow water flows and pollutant diffusions

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An hp‐adaptive discontinuous Galerkin method for shallow water flows

An adaptive spectral/hp discontinuous Galerkin method for the two‐dimensional shallow water equations is presented. The model uses an orthogonal modal basis of arbitrary polynomial order p defined on unstructured, possibly non‐conforming, triangular elements for the spatial discretization. Based on a simple error indicator constructed by the solutions of approximation order p and p?1, we allow both for the mesh size, h, and polynomial approximation order to dynamically change during the simulation. For the h‐type refinement, the parent element is subdivided into four similar sibling elements. The time‐stepping is performed using a third‐order Runge–Kutta scheme. The performance of the hp‐adaptivity is illustrated for several test cases. It is found that for the case of smooth flows, p‐adaptivity is more efficient than h‐adaptivity with respect to degrees of freedom and computational time. Copyright © 2010 John Wiley & Sons, Ltd.     

用离散速度方法计算浅水长波方程

用离散速度法计算浅水波方程，将空气动力学方程和浅水波方程作了比较，用Nadiga提出的近平衡流动方法模拟浅水波方程的连续和间断解。计算了一维的溃坝波问题和Thacker提出的连续解问题，结果与精确解作了比较，并且计算了水流跃过障碍物的问题。     

A dual‐weighted trust‐region adaptive POD 4D‐VAR applied to a finite‐element shallow‐water equations model

We consider a limited‐area finite‐element discretization of the shallow‐water equations model. Our purpose in this paper is to solve an inverse problem for the above model controlling its initial conditions in presence of observations being assimilated in a time interval (window of assimilation). We then attempt to obtain a reduced‐order model of the above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4‐D VAR. Different approaches of POD implementation of the reduced inverse problem are compared, including a dual‐weighed method for snapshot selection coupled with a trust‐region POD approach. Numerical results obtained point to an improved accuracy in all metrics tested when dual‐weighing choice of snapshots is combined with POD adaptivity of the trust‐region type. Results of ad‐hoc adaptivity of the POD 4‐D VAR turn out to yield less accurate results than trust‐region POD when compared with high‐fidelity model. Directions of future research are finally outlined. Copyright © 2009 John Wiley & Sons, Ltd.     

复杂地形条件下WENO-Roe方法在浅水模拟中的改进

在分析浅水方程与二维空气运动方程差异的基础上,在不改变原有浅水方程形式的前提下,提出了局部水位法离散连续方程,并针对动量方程中底坡项提出了更具普遍意义的离散方法,保证了浅水方程离散后的平衡性。通过不规则地形下潮流以及混合流流动的模拟,得到的计算结果符合物理实际,与精确解符合良好,最大相对误差不超过4%,验证了此方法在复杂地形上的平衡性,同时本方法又具有良好的间断捕捉能力与稳定性。     

On the use of sensitivity analysis in model reduction to predict flows for varying inflow conditions

The proper orthogonal decomposition (POD)‐based model reduction method is more and more successfully used in fluid flows. However, the main drawback of this methodology rests in the robustness of these reduced order models (ROMs) beyond the reference at which POD modes have been derived. Any variation in the flow or shape parameters within the ROM fails to predict the correct dynamics of the flow field. To broaden the spectrum of these models, the POD modes should have the global characteristics of the flow field over which the predictions are required. Mixing of snapshots with varying parameters is one way to improve the global nature of the modes but is computationally demanding because it requires full‐order solutions for a number of parameter values in order to assemble atextitrich enough database on which to perform POD. Instead, we have used sensitivity analysis (SA) to include the flow and shape parameters influence during the basis selection process to develop more robust ROMs for varying viscosity (Reynolds number), changing orientation and shape definition of bodies. This study aims at extending these ideas to inflow conditions to demonstrate the effectiveness of the proposed approach in capturing the effect of varying inflow on the dynamics of the flow over an elliptic cylinder. Numerical experiments show that the newly derived models allow for a more accurate representation of the flows when exploring the parameter space. Copyright © 2011 John Wiley & Sons, Ltd.     

Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations

This paper introduces tensorial calculus techniques in the framework of POD to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree p. Such nonlinear terms have an online complexity of , where k is the dimension of POD basis and therefore is independent of full space dimension. However, it is efficient only for quadratic nonlinear terms because for higher nonlinearities, POD model proves to be less time consuming once the POD basis dimension k is increased. Numerical experiments are carried out with a two‐dimensional SWE test problem to compare the performance of tensorial POD, POD, and POD/discrete empirical interpolation method (DEIM). Numerical results show that tensorial POD decreases by 76× the computational cost of the online stage of POD model for configurations using more than 300,000 model variables. The tensorial POD SWE model was only 2 to 8× slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM SWE model to compute its offline stage faster than POD and tensorial POD approaches. Copyright © 2014 John Wiley & Sons, Ltd.     

Efficient identification of uncertain parameters in a large‐scale tidal model of the European continental shelf by proper orthogonal decomposition

The adjoint method can be used to identify uncertain parameters in large‐scale shallow water flow models. This requires the implementation of the adjoint model, which is a large programming effort. The work presented here is inverse modeling based on model reduction using proper orthogonal decomposition (POD). An ensemble of forward model simulations is used to determine the approximation of the covariance matrix of the model variability and the dominant eigenvectors of this matrix are used to define a model subspace. An approximate linear reduced model is obtained by projecting the original model onto this reduced subspace. Compared with the classical variational method, the adjoint of the tangent linear model is replaced by the adjoint of a linear reduced forward model. The minimization process is carried out in reduced subspace and hence reduces the computational costs. In this study, the POD‐based calibration approach has been implemented for the estimation of the depth values and the bottom friction coefficient in a large‐scale shallow sea model of the entire European continental shelf with approximately 10⁶ operational grid points. A number of calibration experiments is performed. The effectiveness of the algorithm is evaluated in terms of the accuracy of the final results as well as the computational costs required to produce these results. The results demonstrate that the POD calibration method with little computational effort and without the implementation of the adjoint code can be used to solve large‐scale inverse shallow water flow problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Concurrent use of finite element and finite volume methods for shallow water flows in locally 1‐D channel networks

A numerical model is developed for shallow water equation in locally 1‐D channel networks. The model concurrently uses the standard Galerkin finite element method for the continuity equation and the finite volume method with an upwind scheme for the momentum equation. The surface gradient method is consistently employed. A minimum treatment is given for channel junctions so that application to multiply connected channels do not require any special consideration The model is capable of computing different types of transcritical flows, wet and dry flows, and flows with complex source terms. Standardized test problems and laboratory experimental data are used for verifying the model. Applicability of the models is validated in a multiply connected channel network draining hydromorphic farmlands located in a West African savanna, and Manning's roughness coefficient is identified, so that the steady solution is consistent with field observations. Unsteady simulation demonstrates that the model is capable of stably reproducing shifts of hydraulic jumps in flows of sub‐millimeter water depths. Copyright © 2011 John Wiley & Sons, Ltd.