Locally Limited and Fully Conserved RKDG2 Shallow Water Solutions with Wetting and Drying

This work extends a well-balanced second-order Runge-Kutta discontinuous Galerkin (RKDG2) scheme to provide conservative simulations for shallow flows involving wetting and drying over irregular topographies with friction effects. For this purpose, a wetting and drying technique designed originally for a finite volume (FV) scheme is improved and implemented, which includes the discretization of friction source terms via a splitting implicit integration approach. Another focus of this work is to design a fully conserved RKDG2 scheme to provide conservative solutions for both mass and momentum through a local slope limiting process. Several steady and transient benchmark tests with/without friction effects are simulated to validate the new solver and demonstrate the effects of different slope limiting processes, i.e. globally and locally slope limiting processes.     

A discontinuous Galerkin algorithm for the two-dimensional shallow water equations

A new high-resolution finite element scheme is introduced for solving the two-dimensional (2D) depth-integrated shallow water equations (SWE) via local plane approximations to the unknowns. Bed topography data are locally approximated in the same way as the flow variables to render an instinctive well-balanced scheme. A finite volume (FV) wetting and drying technique that reconstructs the Riemann states by ensuring non-negative water depth and maintaining well-balanced solution is adjusted and implemented in the current finite element framework. Meanwhile, a local slope-limiting process is applied and those troubled-slope-components are restricted by the minmod FV slope limiter. The inter-cell fluxes are upwinded using the HLLC approximate Riemann solver. Friction forces are separately evaluated via stable implicit discretization to the finite element approximating coefficients. Boundary conditions are derived and reported in details. The present model is validated against several test cases including dam-break flows on regular and irregular domains with flooding and drying.     

A fully implicit wetting–drying method for DG-FEM shallow water models,with an application to the Scheldt Estuary

Resolving the shoreline undulation due to tidal excursion is a crucial part of modelling water flow in estuaries and coastal areas. Nevertheless, maintaining positive water column depth and numerical stability has proved out to be a very difficult task that requires special attention. In this paper we propose a novel wetting–drying method in which the position of the sea bed is allowed to fluctuate in drying areas. The method is implemented in a Discontinuous Galerkin Finite Element Model (DG-FEM). Unlike most methods in the literature our method is compatible with fully implicit time-marching schemes, thus reducing the overall computational cost significantly. Moreover, global and local mass conservation is guaranteed which is crucial for long-term environmental applications. In addition consistency with tracer equation is also ensured. The performance of the proposed method is demonstrated with a set of test cases as well as a real-world application to the Scheldt Estuary. Due to the implicit time integration, the computational cost in the Scheldt application is reduced by two orders of magnitude. Although a DG-FEM implementation is presented here, the wetting–drying method is applicable to a wide variety of shallow water models.     

A Godunov-type scheme for modelling 1D channel flow with varying width and topography

One-dimensional (1D) numerical models have long been used in simulating fluvial hydrodynamics. While most of these models are based on the solutions to some approximate forms of the fully 1D St. Venant equations, it is desirable to have a 1D code that can deal with those highly dynamic and complex flows under certain flood conditions, with full consideration of the convective and source terms. This paper therefore presents a Godunov-type alternative for solving the 1D inhomogeneous shallow water equations with complex source terms. The model is also implemented with a wetting and drying condition to avoid producing negative water depth. The proposed model is validated by a selection of steady and transient hydraulic problems with reference solutions.     

Coupling of two- and three-dimensional hydrodynamic numerical models for simulating wind-induced currents in deep basins

The main interest of the present study is the simulation of wind-induced currents in closed water bodies with shallow and deep regions. This paper describes a low time consumption numerical modelling technique for the simulation of free-surface flow over a geometrically complex bed. To achieve this, a technique employing coupled two- and three-dimensional flow solvers is developed for simulation of the flow. The conjunctive model consists of an upper part 2D Shallow Water Flow Solver (2D-SWFS) coupled with a 3D pseudo-compressible flow solver (3D-PCFS) for the deep regions with a proper interface boundary condition. The 2D-SWFS and 3D-PCFS solvers are coupled via an interfacial shear stress gradient and pressure effects. Time stepping is performed for the 2D solver, and an iterative procedure is employed by the 3D solver to satisfy the equilibrium constraints for the interfacial boundary. The model is able to consider 2D wetting and drying shallow regions without any underlying deep water. Both the 2D and 3D models use nodal based Galerkin finite volume method (GFVM) for solving the governing equations on the unstructured meshes. The accuracy of both models in solving the effective phenomena is examined by comparing the results of simulated test cases with readily available analytical solutions and experimental measurements. Finally, the accuracy of the conjunctive model is assessed by comparing its results for test cases with analytical solutions and experimental measurements from the literature. The new simulation method is then used to solve a wind-induced flow problem in a basin with deep water surrounded by shallow water parts.     

Performance and Comparison of Cell-Centered and Node-Centered Unstructured Finite Volume Discretizations for Shallow Water Free Surface Flows

Finite volume (FV) methods for solving the two-dimensional (2D) nonlinear shallow water equations (NSWE) with source terms on unstructured, mostly triangular, meshes are known for some time now. There are mainly two basic formulations of the FV method: node-centered (NCFV) and cell-centered (CCFV). In the NCFV formulation the finite volumes, used to satisfy the integral form of the equations, are elements of the mesh dual to the computational mesh, while for the CCFV approach the finite volumes are the mesh elements themselves. For both formulations, details are given of the development and application of a second-order well-balanced Godunov-type scheme, developed for the simulation of unsteady 2D flows over arbitrary topography with wetting and drying. The popular approximate Riemann solver of Roe is utilized to compute the numerical fluxes, while second-order spatial accuracy is achieved with a MUSCL-type reconstruction technique. The Green-Gauss (G-G) formulation for gradient computations is implemented for both formulations, in order to maintain a common framework. Two different stencils for the G-G gradient computations in the CCFV formulation are implemented and tested. An edge-based limiting procedure is applied for the control of the total variation of the reconstructed field. This limiting procedure is proved to be effective for the NCFV scheme but inadequate for the CCFV approach. As such, a simple but very effective modification to the reconstruction procedure is introduced that takes into account geometrical characteristics of the computational mesh. In addition, consistent well-balanced second-order discretizations for the topography source term treatment and the wet/dry front treatment are presented for both FV formulations, ensuring absolute mass conservation, along with a stable friction term treatment.     

Numerical solution of tidal currents at marine waterways using wet and dry technique on Galerkin finite volume algorithm

In the present paper the Galerkin finite volume method (FVM) is developed for solution of unsteady; two dimensional shallow water flow equations on unstructured triangular meshes. The numerical model considered two types of wetting and drying process which is an essential technique for modelling the tidal flow in the coastal zones with arbitrary topography. The use of triangular cells facilitates local refinement in the areas with considerable bed elevation variations. Hydrostatic pressure distribution was assumed and the effects of bed slope and friction are considered in two equations of motion. In order to damp out the unwanted numerical oscillations and reduction of instability of the numerical model and its stability during model running, the artificial viscosity was added to the formulation.The quality of the solution result and the accuracy of the applied model is assessed by comparison between numerical results and reported data in the literature for flow in a channel with spur dike which showed an average error of 6% with the experiment data. In addition to assure model accuracy and capability in wetting and drying situations, the current model was examined on an inclined bed with flooding and dewatering process. Numerical Result analysis showed that the second applied wet/dry technique enhanced outcomes from 5% average error to 3%. As a real world case study, Qeshm canal is considered in this study. In the Qeshm canal (located in the east part of the Persian Gulf) the flow pattern is formed by tidal currents via two open boundaries in the west and east ends. Besides, the computed water elevation and velocity at Kaveh port are compared with the available measurements for a period of time, in which the computed water elevation is much closer to the observed data rather than the computed velocity.     

A numerical scheme for solving of shallow water equations — microcomputer version for IBM PC

The implicit four-point scheme for solving unsteady shallow water equations was tested for rapidly varied flow conditions, generated by a hydropower station. A microcomputer version of FORTRAN code was developed and implemented for the family of IBM PC and compativle computers. Results of numerical experiments were compared with observed water level elevations recorded during one weekly period during operation of the hydropower facility.     

A positivity-preserving zero-inertia model for flood simulation

Zero-Inertia Models (ZIMs), or Diffusion-Wave Models (DWMs), have been widely used in flood modelling in the last decade. In this work, an alternative formulation is proposed based on a new depth-positivity-preserving condition to solve the zero-inertia governing equation. The new condition does not use a flux limiter and is practical for flood simulations with wetting and drying over complex domain topographies. Two time stepping methods are considered and studied along with the proposed numerical model. The first one is based on the Courant-Friedrichs-Lewy (CFL) condition, which is widely used to control the time step for the explicit shallow water equation solvers; the second one is the adaptive time stepping (ATS) reported by Hunter et al. [1], which was specifically designed for a DWM. Numerical results and root-mean-square-error (RMSE) analysis show that the new model is able to provide stable and accurate solutions without the necessity for a flux limiter. Computational efficiency is significantly improved under the CFL constraint.     

An efficient unstructured MUSCL scheme for solving the 2D shallow water equations

The aim of this paper is to present a novel monotone upstream scheme for conservation law (MUSCL) on unstructured grids. The novel edge-based MUSCL scheme is devised to construct the required values at the midpoint of cell edges in a more straightforward and effective way compared to other conventional approaches, by making better use of the geometrical property of the triangular grids. The scheme is incorporated into a two-dimensional (2D) cell-centered Godunov-type finite volume model as proposed in Hou et al. (2013a,c) to solve the shallow water equations (SWEs). The MUSCL scheme renders the model to preserve the well-balanced property and achieve high accuracy and efficiency for shallow flow simulations over uneven terrains. Furthermore, the scheme is directly applicable to all triangular grids. Application to several numerical experiments verifies the efficiency and robustness of the current new MUSCL scheme.     

A Runge-Kutta discontinuous Galerkin method for the Euler equations

This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for the Euler equations of gas dynamics from the viewpoint of kinetic theory. Like the traditional gas-kinetic schemes, our proposed RKDG method does not need to use the characteristic decomposition or the Riemann solver in computing the numerical flux at the surface of the finite elements. The integral term containing the non-linear flux can be computed exactly at the microscopic level. A limiting procedure is carefully designed to suppress numerical oscillations. It is demonstrated by the numerical experiments that the proposed RKDG methods give higher resolution in solving problems with smooth solutions. Moreover, shock and contact discontinuities can be well captured by using the proposed methods.     

Transport of pollutant in shallow flows: A space–time CE/SE scheme

This paper focuses on implementation of space–time CE/SE scheme for computing the transport of a passive pollutant by a flow. The flow model comprises of the Saint-Venant system of shallow water equations and the pollutant propagation is described by a transport equation. The one-dimensional and two-dimensional flow equations are numerically investigated using the CE/SE scheme. A number of test problems are presented to check the accuracy and efficiency of the proposed scheme. The results of CE/SE scheme are compared with the central scheme. Both the schemes are found to be in close agreement. However, our proposed CE/SE scheme accurately captures shocks and discontinuous profiles.     

Modeling of wetting–drying transitions in free surface flows over complex topographies

Numerical techniques development for the modeling and simulation of free surface flows has generated great interests over the last decades. In hydraulic engineering, the objectives include the predictions of dam break waves' propagation, fluvial floods and other catastrophic flooding phenomenon, the modeling of estuarine and coastal circulations, and the design and optimization of hydraulic structures. Most of the flooding events involve wetting and drying lands which are critical for the numerical modeling, especially when dealing with complex topographies. Extreme slopes and abrupt changes in irregular geometries, have often led to significant numerical errors and stability difficulties, and these are more critical for propagations over complex dry beds. This paper presents a simple and efficient numerical model for the wetting and drying effects over complex bathymetries. An overview of the key methods that have been suggested since the pioneering studies is first presented. A 2-D cell-centered finite-volume scheme is then proposed for solving the shallow-water equations using both structured and unstructured fixed meshes. Steady state C-property and global mass conservation properties are satisfied using appropriate numerical fluxes and wet/dry interfaces treatments. The resulting numerical model proved stable and robust and was validated through some benchmarks tests, including comparisons with exact solutions and experimental data, and a real case of wetting–drying simulation in a portion of the river “Rivière des Prairies” in a suburb of Laval, Quebec.     

A local discontinuous Galerkin method for a doubly nonlinear diffusion equation arising in shallow water modeling

In this paper, we study a local discontinuous Galerkin (LDG) method to approximate solutions of a doubly nonlinear diffusion equation, known in the literature as the diffusive wave approximation of the shallow water equations (DSW). This equation arises in shallow water flow models when special assumptions are used to simplify the shallow water equations and contains as particular cases: the Porous Medium equation and the parabolic p-Laplacian. Continuous in time a priori error estimates are established between the approximate solutions obtained using the proposed LDG method and weak solutions to the DSW equation under physically consistent assumptions. The results of numerical experiments in 2D are presented to verify the numerical accuracy of the method, and to show the qualitative properties of water flow captured by the DSW equation, when used as a model to simulate an idealized dam break problem with vegetation.     

High-Order Well-Balanced Finite Difference WENO Schemes for a Class of Hyperbolic Systems with Source Terms

In this paper, we generalize the high order well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, designed earlier by us in Xing and Shu (2005, J. Comput. phys. 208, 206–227) for the shallow water equations, to solve a wider class of hyperbolic systems with separable source terms including the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. Properties of the scheme for the shallow water equations (Xing and Shu 2005, J. Comput. phys. 208, 206–227), such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions, are maintained for the scheme when applied to this general class of hyperbolic systems     

Optimal control of water distribution systems by network flow theory

This paper presents a modeling technique and an optimal control scheme for water distribution networks. To overcome the large scale and nonlinearity of the network, a network aggregation method and a two-level control scheme are developed. The first level of the scheme decides operating points using a nonlinear optimization method, where the pressure/flow equations are solved using a high-speed technique derived from network flow theory. The second level is a feedback control around the operating points, which absorbs estimate error and small variations in consumption. The scheme has been implemented on a minicomputer system and is presently in operation.     

Finite-Volume-Particle Methods for Models of Transport of Pollutant in Shallow Water

We present a new hybrid numerical method for computing the transport of a passive pollutant by a flow. The flow is modeled by the Saint-Venant system of shallow water equations and the pollutant propagation is described by a transport equation. The idea behind the new finite-volume-particle (FVP) method is to use different schemes for the flow and the pollution computations: the shallow water equations are numerically integrated using a finite-volume scheme, while the transport equation is solved by a particle method. This way the specific advantages of each scheme are utilized at the right place. This results in a significantly enhanced resolution of the computed solution     

Application of a discontinuous Galerkin finite element method to special relativistic hydrodynamic models

A Runge–Kutta discontinuous Galerkin (RKDG) finite element method is proposed for solving the special relativistic hydrodynamic (SRHD) equations and as a limiting case the ultra-relativistic hydrodynamic (URHD) equations. The latter model is obtained by ignoring the rest-mass energy when the internal energy of fluid particles is sufficiently large. Several test problems of SRHD and URHD models are carried out. For validation, the results of DG-method are compared with the staggered central scheme. The numerical results verify the accuracy of the proposed method qualitatively and quantitatively.     

Implementation of a discontinuous Galerkin morphological model on two-dimensional unstructured meshes

The shallow water equations are used to model large-scale surface flow in the ocean, coastal rivers, estuaries, salt marshes, bays, and channels. They can describe tidal flows as well as storm surges associated with extreme storm events, such as hurricanes. The resulting currents can transport bed load and suspended sediment and result in morphological changes to the seabed. Modeling these processes requires tightly coupling a bed morphology equation to the shallow water equations. Discontinuous Galerkin finite element methods are a natural choice for modeling this coupled system, given the need to solve these problems on unstructured computational meshes, as well as the desire to implement hp-adaptivity for capturing the dynamic features of the solution. However, because of the presence of non-conservative products in the momentum equations, the standard DG method cannot be applied in a straightforward manner. To rectify this situation, we summarize and follow an extended approach described by Rhebergen et al., which uses theoretical results due to Dal Maso et al. appearing in earlier work. In this paper, we focus on aspects of the implementation of the morphological model for bed evolution within the Advanced Circulation (ADCIRC) modeling framework, as well as the verification of the RKDG method in both h (mesh spacing) and p (polynomial order). This morphological model is applied to a number of coastal engineering problems, and numerical results are presented, with attention paid to the effects of h- and p-refinement in these applications. In particular, it is observed that for sediment transport, piecewise constant (i.e., finite volume) approximations of the bed are very over-diffusive and lead to poor sediment solutions.     

基于浅水波理论液体晃动初值问题的数值模拟

针对日益受到关注的液体晃动问题,提出了一种基于浅水波理论的研究方案.该方案采用浅水波理论而非势流理论导出系统控制方程,并通过哈密顿体系表达;利用中心有限差分法和Stormer-Verlet算法进行空间和时间离散;模拟了不同初值条件下的液体晃动情况并对比分析了影响系统非线性响应的主要因素.结果表明,基于浅水波理论能有效解决液体晃动问题;与Euler格式对比,Stormer-Verlet算法精度较高;除共振外对于系统非线性响应的影响容器初始位移比初始速度更显著;非共振情况一定条件下,充液容器运动过程中液体晃动能起到阻尼作用.