Numerical Methods for Shallow Water Equations and Related Models

Title: About fully-well-balanced schemes for shallow-water equations

Speaker: Christophe Berthon, University of Nantes, France.

Abstract: The present work concerns the numerical approximation of the weak solutions of the well-known shallow-water model. A particular attention is paid on the steady states. Indeed such specific solutions are essential to ensure the accuracy of the scheme when considering some important regimes. A large literature is devoted to numerical schemes able to exactly preserve the so-called lake at rest which coincides to the simpler (linear) stationary regime. More recently, the nonlinear steady solutions, governed by the Bernouilli's equations, have been considered. The situation turns out to be drastically distinct because of the strong nonlinearities. In the present talk, we present several approach, based on Godunov-type methods, to deal with this severe problem. In addition, we present applications coming from nonlinear friction source term models. A MUSCL second-order extension is also proposed. This talk is illustrated with several numerical experiments.

Title: Shallow water hydro-sediment-morphodynamic models - advances and challenges

Speaker: Zhixian Cao, State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, China

Abstract: Shallow water flow often evokes active sediment transport and changes in bed elevation and composition, which in turn conspire to modify the flow. The interactive processes of flow, sediment transport and bed evolution constitute a hierarchy of physical problems of significant interest in the fields of water resources engineering, fluvial and coastal geomorphology, flood risk management as well as environmental and ecological wellbeing in surface waters. Shallow water hydro-sediment-morphodynamic (SHSM) models have been increasingly widely applied to enhance the understanding of such processes over the last several decades. Here the recent advances in SHSM models are addressed along with the challenges we are facing. First, the traditional fully coupled, depth-averaged SHSM model is briefed, based on the fundamental mass and momentum conservation laws for the quasi-single-phase flow of the water-sediment mixture. Second, the extended version is outlined as per a double layer-averaged SHSH model, which facilitates the resolution of sharply stratified sediment-laden flow, i.e., a clear water flow over a sediment-laden flow. Then, a depth-averaged two-phase SHSM model is presented, in which the inter-phase and inter-particle interactions are explicitly incorporated. Typical applications of these models are presented, including erodible-bed dam-break floods, turbidity currents and landslide-generated waves in reservoirs, navigational waterway evolution, and debris flows. Finally, challenges are discussed for wider applications of SHSM models.

Title: A New Hydrostatic Reconstruction Scheme For Shallow Water Equations Based on Subcell Reconstructions

Speaker: Guoxian Chen, Wuhan University, China

Abstract: A key difficulty in the analysis and numerical approximation of the shallow water equations is the non-conservative product of measures due to the gravitational force acting on a sloped bottom. Solutions may be non-unique, and numerical schemes are not only consistent discretizations of the shallow water equations, but they also make a decision how to model the physics. Our derivation is based on a subcell reconstruction using infinitesimal singular layers at the cell boundaries, as inspired by [Noelle, Xing, Shu, JCP 2007]. One key step is to separate the singular measures. Another aspect is the reconstruction of the solution variables in the singular layers. We study three reconstructions. The first leads to the well-known scheme of [Audusse, Bristeau, Bouchut, Klein, Perthame, SISC 2004], which introduces the hydrostatic reconstruction. The second is a modification proposed in [Morales, Castro, Pares, AMC 2013], which analyzes if a wave has enough energy to overcome a step. The third is our new scheme, and borrows its structure from the wet-dry front. For a number of cases discussed in recent years, where water runs down a hill, Audusse’s scheme converges slowly or fails. Morales’ scheme gives a visible improvement. Both schemes are clearly outperformed by our new scheme.

Title: Structure Preserving Numerical Methods for Hyperbolic Systems of Conservation and Balance Laws

Speaker: Alina Chertock, North Carolina State University, USA

Abstract: Shallow water and related models are widely used as a mathematical framework to study water flows in rivers and coastal areas as well as to investigate a variety of phenomena in atmospheric sciences and oceanography. These models are governed by a system of balance laws, which can generate solutions with a complex wave structure including nonlinear shock and rarefaction waves, as well as linear contact waves that may appear in the case of discontinuous bottom topography. The level of complexity may increase even further when solutions of the hyperbolic system includes additional terms such as friction terms, geometrical terms, nonconservative products, etc., which are needed to be taken into account in order to achieve a proper description of the studied physical phenomena. In such cases, it is extremely important to design a numerical method that is not only consistent with the given PDEs, but also preserves certain structural and asymptotic properties of the underlying problem at the discrete level. While a variety of numerical methods for such models have been successfully developed, there are still many open problems, for which the derivation of reliable high-resolution numerical methods still remains to be an extremely challenging task.

In this talk, I will discuss recent advances in the development of two classes of structure preserving numerical methods for nonlinear hyperbolic systems of conservation and balance laws. In particular, I will present well-balanced and positivity preserving numerical schemes, that is, the methods which are capable of exactly preserving some steady-state solutions as well as maintaining the positivity of the numerical quantities when it is required by the physical application.

Title: Numerical methods and numerical simulation in Oceanography

Speaker: Wai-Sun Don, School of Mathematical Sciences, Ocean University of China, China

Abstract: We will present a brief overview of the development of physical oceanography and a brief introduction of Key Laboratory of Physical Oceanography (POL) at the Ocean University of China (OUC), known for its extensive research in oceanography and related fields, such as physical oceanography, meteorology, atmospheric physics and environment.  The coastal circulation and mass transport, especially the numerical simulations and numerical assimilations are two of many major research directions here at OUC.  In the inversion study of open boundary conditions, the dynamic constraints and observations are considered as a whole, and the deviations between the numerical results and the observations are used as the external forces to drive the adjoint equations, so that the estimates of the appropriate boundary conditions are obtained. The method does not require information about the boundary conditions, but the satellite altimeter observation data and the tide station data to retrieve the open boundary conditions.  Furthermore, the high-resolution numerical algorithm and structure-preserving algorithm are applied to the simulation. For example, WENO scheme can guarantee essentially non-oscillatory at high gradients. Hybrid schemes are used to reduce the calculation time and dispersion and dissipation errors. Discontinuous Galerkin (DG) method is a class of finite element methods using discontinuous piecewise polynomial space as the solution and test function spaces. It has been used extensively in solving the shallow water equations. The well-balanced schemes are designed to preserve exactly the steady-state solution up to the machine error with relatively coarse meshes.  Some results from several classical examples will be presented in the talk..

Title: Numerical simulation of urban flooding with high-resolution topography using CPU and GPU parallel computations

Speaker: Xu Dong, State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, China

Abstract: Flooding cause by dam-break, or tsunami caused are always great threats to coastal or riverside areas, where highly urbanized modern cities are usually located. The fatalness of such floods has two aspects: Firstly, the overwhelming power of tsunami or dam-break waves are extremely destructive for coastal engineering structures and buildings; secondly, the fast propagating flooding waves leave quite short time to emergency responses. Therefore, it is extremely important to correctly predict flooding wave propagation and potential urban flooding as quickly as possible. In this paper, parallel computing techniques using both MPI (Message Passing Interface) and GPU CUDA was adopted to accelerate the simulation of dam-break flows. Results provide strong evidences of its applicability in the simulation of tsunami flood flow in urban areas with dense buildings.

Mathematical model was established for dam-break flow based on finite volume method. Godunov scheme was adopted to evaluate the flux of the Riemann problem. Spatial decomposition using blocks was used for large scale parallelization and computation acceleration. With these, fine simulation of the flood propagation in urban cities was realized. Numerical simulation results show that: buildings undergoing the peak hydrodynamic load at the initial stage of the arrival of the dam-break wave front; Owing to the wave dissipation and shielding effects by the upstream buildings, the load decreases with the increase of the distance to the dam-break gate; the existence of buildings slows down the propagation of the flood. Parallel computation tests show that the buffered data mode is more suitable for large scale parallel computation of shallow water equations than the non-buffered mode. Large scale parallel computation with current technique can fulfil real-time simulation of flood propagation in urban areas as large as 400 km2, which can scientifically support the fast decision and emergency evacuation when riverside or coastal cities experience flood threats from dam-break or tsunamis.  

Figure 1  Free surface of dam-break flooding in urban areas with buildings

Title: Modified shallow-water equations for direct bathymetry reconstruction

Abstract: Improved capabilities of remote sensing and sea surface elevation data availability allow to obtain high-resolution elevation maps for rivers, estuaries, coastal seas, etc. However, the availability of bathymetry information is much more scarce in many locations of interest, and data collection methods for large area coverages are either inaccurate or expensive. To address this problem, a modified shallow water system is proposed that uses bathymetry as the primary unknown, whereas the free surface elevation field enters the system as a prescribed quantity. In addition to formulating and discretizing this modified system, our talk demonstrates first numerical results.

Title: Central-Upwind Schemes for Shallow Water Models

Speaker: Alexander Kurganov, Southern University of Science and Technology, China

Abstract: In the first part of the talk, I will describe a general framework for designing finite-volume methods (both upwind and central) for hyperbolic systems of conservation laws. I will focus on Riemann-problem-solver-free non-oscillatory central schemes and, in particular, on central-upwind schemes that belong to the class of central schemes, but has some upwind features that help to reduce the amount of numerical diffusion typically present in staggered central schemes such as, for example, the first-order Lax-Friedrichs and second-order Nessyahu-Tadmor scheme. 
In the second part of the talk, I will discuss how central-upwind schemes can be extended to hyperbolic systems of balance laws, such as the Saint-Venant system and related shallow water models. The main difficulty in this extension is preserving a delicate balance between the flux and source terms. This is especially important in many practical situations, in which the solutions to be captured are (relatively) small perturbations of steady-state solutions. The other crucial point is preserving positivity of the computed water depth (and/or other quantities, which are supposed to remain nonnegative). I will present a general approach of designing well-balanced positivity preserving central-upwind schemes and illustrate their performance on a number of shallow water models.

Title: A High-Performance Integrated Hydrodynamic Modelling System for Real-Time Flood Forecasting

Speaker: Qiuhua Liang, Newcastle University, UK

Abstract: Reliable prediction of flash floods induced by intense rainfall is beyond the capability of traditional hydrological and simplified hydraulic models due to their inability to representing highly transient flood hydrodynamics over complex topographies. The fully hydrodynamic models based on numerical solution to the shallow water equations (SWEs), especially those developed using a shock-capturing numerical scheme, represent the current state-of-the-art in flash flood modelling. However, these full hydrodynamic models are commonly computationally demanding, restricting their application to large-scale simulations across an entire city or catchment. Harnessing the recent GPU parallel computing technologies, a High-Performance Integrated hydrodynamic Modelling System (HiPIMS) has been recently developed at Newcastle. HiPIMS numerically solves the SWEs using a finite volume Godunov-type shock-capturing scheme. An innovative surface reconstructed method (SRM) and a fully implicit scheme are implemented to respectively discretise the slope source terms and friction source terms, ensuring 'well-balanced solutions' even at the condition of disappearing water depth. HiPIMS achieves multi-GPU and multi-system (machine) parallelisation through the NVIDIA CUDA parallel computing platform and MPI (Message Passing Interface). To support real-time flood forecasting, computing environment/ interface has also been developed to automatically convert the numerical weather forecast products (generated by the UKV model) from the UK Met Office to drive HiPIMS to simulate the resulting flooding process. This new high-resolution hydrodynamic flood forecasting system was then applied to reproduce the 2012 Newcastle flash flood event over a 400km2 urban area at an unprecedented 2m resolution and the 2015 floods caused by Storm Desmond over the 2500km2 Eden Catchment at a 5m resolution; both simulations were run faster than real time, effectively demonstrating HiPIMS's potential for operational real-time flood forecasting.

Title: A physics-based morphological model for simulating the evolution processes of braided channels

Speaker: Binliang Lin, Department of Hydraulic Engineering, Tsinghua University, China

Abstract: This talk presents the development and application of a physics-based morphological model for simulating the evolution processes of braided channels. This model comprises a shallow water flow sub-model for both the gradually and rapidly varying unsteady flows and a bed load sediment transport sub-model for non-uniform sediments. The sheltering effects of non-uniform particles and the lateral sediment transport due to bed slope and secondary flow are taken into account in the sediment sub-model. Channel bed level change is calculated according to the erosion/deposition rate, and the bank movement is modelled according to the submerged angle of repose, which is valid for braided rivers in natural and experimental conditions. The model has been applied to braided rivers produced in flume experiments, and the numerical model-predicted channel patterns are shown to generally well resemble the experimental rivers. Most of the morphodynamic processes observed in the experiments can be found in the numerical predictions, including the evolution of the channel from a single straight channel to a multi-thread pattern and local morphologic changes. The mechanisms of the morphodynamic evolution of multi-thread flows are investigated, in which the process of grain sorting occurs under the interaction of fluid and sand, and there effects on channel migration are discussed.

Title: Well-balanced positivity preserving central-upwind scheme with a novel wet/dry reconstruction on triangular grids for the Saint-Venant system

Speaker: Xin Liu, Southern University of Science and Technology, China

Abstract: We construct a well-balanced positivity preserving central-upwind scheme for the two-dimensional Saint-Venant system of shallow water equations. As in [Bryson et al., M2AN Math. Model. Numer. Anal., 45 (2011), 423–446], our scheme is based on a continuous piecewise linear discretization of the bottom topography over an unstructured triangular grid. The main new technique is a special reconstruction of the water surface in partially flooded cells. This reconstruction is an extension of the one-dimensional wet/dry reconstruction from [Bollermann et al., J. Sci. Comput., 56 (2013), 267–290]. The positivity of the computed water depth is enforced using the “draining” time-step technique introduced in [Bollermann et al., Commun. Comput. Phys., 10 (2011), 371–404]. The performance of the proposed central-upwind scheme is tested on a number of numerical experiments.

Title: Asymptotic preserving schemes for low Froude number shallow water flows

Speaker: Maria Lukacova, University of Mainz, Germany

Abstract: We present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically arise in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by central finite difference method. On the other hand, the rest nonlinear advection part will be approximated explicitly in time and in space by means of any standard finite volume scheme. Time integration is realized by the implicit-explicit (IMEX) method. For low Froude number flows we prove asymptotic preserving stability and consistency of the resulting scheme. If time permits we present uniform (asymptotic preserving) error estimates for a related model of viscous isentropic Navier-Stokes equations in low Mach number limit.

The present research has been partially supported by the German Science Foundation (DFG) under the Collaborative Research Centers TRR 146 and TRR 165.

Title: Modeling of shallow flows with nonuniform density and suspended and bed load

Speaker: Majid Mohammadian, University of Ottawa, Canada

Abstract: Shallow water flows with variable density and/or sediment transport are ubiquitous. For example, when a cold mountain stream reaches a larger river or a lake, the difference between the densities of the two system becomes important and can affect the flow. Therefore, the difference in density needs to be considered in the equations to obtain accurate results. Another application is when surface outfalls of desalination or industrial plants enter rivers or coastal waters. Two-dimensional shallow water equations are more suitable than 3-D equations because of their computational efficiency. This is particularly the case in the initial stages of the design where several trial and errors are needed to develop an optimal design.  However, standard shallow water equations do not consider density differences and sediment transport and modified forms of them should be considered.  In the present research, we apply and extend the method proposed by Bryson et al. (2010) to variable density shallow water equations. This scheme offers well balanced property as well as positivity-preserving both for water depth and density. Several test cases demonstrate the performance of the scheme. 

Title: A simple and efficient WENO method for hyperbolic conservation laws

Speaker: Jianxian Qiu, School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, China

Abstract: In this presentation, we present a simple high order weighted essentially non- oscillatory (WENO) schemes to solve hyperbolic conservation laws. The main advantages of these schemes presented in the paper are their compactness, robustness and could maintain good convergence property for solving steady state problems. Comparing with the classical WENO schemes by {G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), 202-228}, there are two major advantages of the new WENO schemes. The first, the associated optimal linear weights are independent on topological structure of meshes, can be any positive numbers with only requirement that their summation equals to one, and the second is that the new scheme is more compact and efficient than the scheme by Jiang and Shu. Extensive numerical results are provided to illustrate the good performance of these new WENO schemes.

Title: 2007-2017: a decade of residual distribution for shallow water flows

Title: Depth averaged Euler system with a given velocity profile

Speaker: Jacques Saint-Marie, CEREMA, INRIA Paris, France

The shallow water equations (SWEs) are built upon the assumption that the horizontal velocity does not vary (or slightly varies) from the bottom to the free surface of the flow. In other words, the horizontal velocities do not depend on the vertical coordinate.

During this talk, we present the origins, the derivation and the properties of a family of models based on the assumption that the horizontal velocities depend on the vertical coordinate through a prescribed shape function. These reduced complexity models are adapted to situations where the shallow water assumption is no more valid, typically traveling waves with short wavelength.

The behavior of the proposed models is confronted with

1. analytical solutions of the incompressible Euler system with free surface,

2. experimental data.

Title: High Order Discontinuous Galerkin Methods for Hyperbolic Conservation Laws with Source Terms

Speaker: Yulong Xing, Ohio State University, USA

Abstract: Shallow water equations with a non-flat bottom topography, and Euler equations under gravitational fields, are two prototype hyperbolic balance laws, with various applications in differential fields. In this presentation, we will talk about high order well-balanced discontinuous Galerkin finite element methods, which can exactly capture the

non-trivial steady state solutions of these models. Some numerical tests are provided to verify the well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.