Method For Shallow Water Equations Research Articles

Combining a Central Scheme with the Subtraction Method for Shallow Water Equations

We present a new numerical scheme that is a well-balanced and second-order accurate for systems of shallow water equations (SWEs) with variable bathymetry. We extend in this paper the subtraction method (resulting in well-balancing) to the case of unstaggered central finite volume methods that computes the numerical solution on a single grid. In addition, the proposed scheme avoids solving Riemann problems occurring at cell boundaries as it employs intermediately a layer of ghost-staggered cells. The proposed numerical scheme is then implemented and validated. We successfully manage to solve classical SWE problems from the literature featuring steady states and other equilibria. The results of the study are consistent with previous research, which supports the use of the proposed method to solve SWEs.

Water-balanced inlet and outlet boundary conditions of the lattice Boltzmann method for shallow water equations

Conservation of mass or water balance is important for the simulation of shallow water projects. The traditional treatment of discharge and water level boundary conditions ignores it in the lattice Boltzmann model for shallow water equations (LABSWE). In this paper, we introduced the water balance equation at these boundary conditions and proposed a universal treatment for the straight boundary of any direction. Six schemes on the unknown particle distribution functions (PDFs) at boundary lattices for given macroscopic quantities (h,u,v) were compared in terms of accuracy and stability, and the nonequilibrium extrapolation (NEQ) scheme and full link nonequilibrium extrapolation (Ful-NEQ) scheme were finally adopted in the proposed treatment. The numerical tests show that the proposed treatment fully satisfies the specified boundary conditions and conservation of mass. Moreover, the distribution of the velocities at the boundaries is consistent along downstream which can reduce the artificial influence length of the boundary.

A consistent mass-conserving C-staggered method for shallow water equations on global reduced grids

The increasing number of cores in modern supercomputers motivated the search for methods with good scalability to model global problems in geophysical fluid dynamics, including the representation of the atmospheric dynamics in global weather forecasting and climate simulations. Grid-point schemes with quasi-uniform spherical grids, along with explicit time integration, are one of the known options for recent dynamical core developments. These methods avoid the concentration of points in the poles and do not require global communication, intending to contribute to the efficiency of parallel implementations. On the other hand, they may yield low accuracy or present errors stimulated by the grid. Here, we target reduced latitude-longitude global grids and build from derivations of previous schemes to propose a method for the shallow water equations on a C-staggered global reduced grid using a combination of finite differences, finite volumes, and explicit time-stepping. This scheme employs only local approximations, has some conservative properties, and uses numerically consistent operators everywhere. Classical and recently proposed benchmarks were used to evaluate convergence, stability, and accumulation of errors. Results indicated that the proposed scheme provides adequate accuracy at competitive computational cost.

An Oscillation-free Discontinuous Galerkin Method for Shallow Water Equations

An Oscillation-free Discontinuous Galerkin Method for Shallow Water Equations

Macroscopic Lattice Boltzmann Method for Shallow Water Equations

The lattice Boltzmann method (LBM) is characterised by its simplicity, parallel processing and easy treatment of boundary conditions. It has become an alternative powerful numerical method in computational physics, playing a more and more important role in solving challenging problems in science and engineering. In particular, the lattice Boltzmann method with the single relaxation time (SLBM) is the simplest and most popular form of the LBM that is used in research and applications. However, there are two long-term unresolved problems that prevent the SLBM from being an automatic simulator for any flows: (1) stability problem associated with the single relaxation time and (2) no method of direct implementation of physical variables as boundary conditions. Recently, the author has proposed the macroscopic lattice Boltzmann method (MacLAB) to solve the Navier–Stokes equations for fluid flows, resolving the aforementioned problems; it is unconditionally stable and uses physical variables as boundary conditions at lower computational cost compared to conventional LBMs. The MacLAB relies on one fundamental parameter of lattice size δx, and is a minimal version of the lattice Boltzmann method. In this paper, the idea of the MacLAB is further developed to formulate a macroscopic lattice Boltzmann method for shallow water equations (MacLABSWE). It inherits all the advantages from both the MacLAB and the conventional LBM. The MacLABSWE is developed regardless of the single relaxation time τ. Physical variables such as water depth and velocity can directly be used as boundary conditions, retaining their initial values for Dirichlet’s boundary conditions without updating them at each time step. This makes not only the model to achieve the exact no-slip boundary condition but also the model’s efficiency superior to the most efficient bounce-back scheme for approximate no-slip boundary condition in the LBMs, although the scheme can similarly be implemented in the proposed model when it is necessary. The MacLABSWE is applied to simulate a 1D unsteady tidal flow, a 2D steady wind-driven flow in a dish-shaped lake and a 2D steady complex flow over a bump. The results are compared with available analytical solutions and other numerical studies, demonstrating the potential and accuracy of the model.

A Consistent Mass-Conserving C-Staggered Method for Shallow Water Equations on Global Reduced Grids

A Consistent Mass-Conserving C-Staggered Method for Shallow Water Equations on Global Reduced Grids

Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Shallow Water Equations

A stochastic Galerkin formulation for a stochastic system of balanced or conservation laws may fail to preserve the hyperbolicity of the original system. In this work, we develop a hyperbolicity-preserving stochastic Galerkin formulation for the one-dimensional shallow water equations by carefully selecting the polynomial chaos expansion of the nonlinear $q^2/h$ term in terms of the polynomial chaos expansions of the conserved variables. In addition, in an arbitrary finite stochastic dimension, we establish a sufficient condition to guarantee the hyperbolicity of the stochastic Galerkin system through a finite number of conditions at stochastic quadrature points. Further, we develop a well-balanced central-upwind scheme for the stochastic shallow water model and derive the associated hyperbolicity-preserving CFL-type condition. The performance of the developed method is illustrated on a number of challenging numerical tests.

An $h$-Box Method for Shallow Water Equations Including Barriers

The shallow water equations provide the basic modeling equations for a number of coastal flooding hazards, such as tsunamis and storm surge. In realistic scenarios, there are often structures important to these flows that have a large extent but small width, including sea walls, berms, and harbor barriers. Explicit time stepping schemes, most often used for the shallow water equations, can then suffer from time step restrictions due to the CFL condition. In this article, we introduce an approach that side-steps these issues by allowing a barrier to have zero-width and to cut a cell arbitrarily without suffering from CFL restrictions. This is done by supplementing existing Riemann solvers and leveraging $h$-box and large time stepping methods. These methods preserve the properties of the Riemann solver and add negligible cost to the original solvers.

High-Order WENO Schemes with an Immersed Boundary Method for Shallow Water Equations on the Tsunami Mitigation with Configurations of Cylinder Array

In this paper, we investigated the flow around vertical circular cylinders with the numerical simulation of shallow water equations. Since tsunami tidal waves have large potential energy due to height difference, the man-made structures near coast are heavily damaged during the propagation of waves. For the damage mitigation of natural disaster, a careful configuration of such obstructive structures is worthy of attention, and it is of great practical importance to provide numerical simulation based on the ideal models of natural or artificial structures in a coastal area. The simplified numerical models are listed as the single large cylinder and four small cylinders of double inline, double staggered, and single four arrangement with fixing the same blockage ratio. Among various up-to-date schemes, the fifth-order WENO scheme is selected with a ghost-cell immersed boundary method, which enabled the easy detection of the arbitrary body shapes emerged in background Cartesian grids. In addition, we proposed the non-reflecting characteristic boundary condition based on the characteristic compatibility equations derived in shallow water equations to suppress the unexpected reflection waves at the open boundary. The analysis on a benchmark problem on solitary wave reflection led to an expected validation with the present numerical model. Finally, reflection and diffraction of a tsunami wave are analyzed against a single and four cylinders of the same blockage density, and it is believed to reduce 20.80% points for the best case of single four cylinders in the view of dissipation ratio defined by the kinematic power.

Study on vegetation flow along Kinta River by using LABSWE™

Turbidity is considered a classical problem in Malaysian rivers. The level of turbidity is dependent on the amount of sediments and biological matter contained in the water. Therefore, this study proposes an approach to turbidity mitigation that involves simulating the effect of density and diameter of vegetation on water flow. This paper presents the numerical modelling of vegetation effect on water flow using Lattice Boltzmann Method for Shallow Water Equations (LABSWE™) integrated with Rigid Vegetation Modelling (RVM). The proposed model was implemented on the Kinta River in Perak state, Malaysia as a case study. The simulation results show the effect of density and diameter of vegetation on water flow velocity. The optimum vegetation density and vegetation diameter values for river water turbidity reduction were found to be 0.05 and 0.005, respectively.

Modelling flow in a fishway using LABSWE™

Fishways are designed to provide a canal for fishes to migrate upstream and downstream when there is a presence of natural or man-made obstructions. In designing the fishway, correct prediction and simulation of the flows play a crucial role. A well designed and constructed fishway will provide a path that allow fish to continue migrating past the obstructions. This paper presents numerical modelling of flows in fishway using Lattice Boltzmann Method for Shallow Water Equations (LABSWE™) in order to demonstrate its simple applicability and accuracy in solving complex flow problems. Lattice Boltzmann Method (LBM) is an alternative numerical method in solving complex fluid flows problems. LBM was found to be efficient and accurate after series complex shallow water problems have been solved. The flow is simulated by using Bhatnagar-Gross-Kook (BGK) collision term. Lattice Boltzmann equation was derived based on shallow water equations. A D2Q9 model was implemented for a more accurate result. The BGK-LBM equations was coded up in MATLAB and the numerical results yield from LBM were then compared with the numerical results solved using Finite Volume Method (FVM). The result comparison was done on velocity vectors, velocity contour and depth contour. From the result comparison, there is good agreement between LBM and FVM which shows that LBM can mimic the flow perfectly. In fact, LBM yield a more detailed and accurate results.

Well-Balanced Discontinuous Galerkin Method for Shallow Water Equations with Constant Subtraction Techniques on Unstructured Meshes

The classical Saint–Venant shallow water equations on complex geometries have wide applications in many areas including coastal engineering and atmospheric modeling. The main numerical challenge in simulating Saint–Venant equations is to maintain the high order of accuracy and well-balanced property simultaneously. In this paper, we propose a high-order accurate and well-balanced discontinuous Galerkin (DG) method on two dimensional (2D) unstructured meshes for the Saint–Venant shallow water equations. The technique used to maintain well-balanced property is called constant subtraction and proposed in Yang et al. (J Sci Comput 63:678–698, 2015). Hierarchical reconstruction limiter with a remainder correction technique is introduced to control numerical oscillations. Numerical examples with smooth and discontinuous solutions are provided to demonstrate the performance of our proposed DG methods.

A stabilized well-balanced RBF-FD meshless method for shallow-water equations

In this work, we are concerned with the study and computing of stabilized radial basis function-generated finite difference (RBF-FD) approximations for shallow-water equations. In order to obtain both stable and highly accurate numerical approximations of convection-dominated shallow-water equations, we use stabilized flat Gaussians (RBFSGA-FD) and polyharmonic splines with supplementary polynomials (RBFPHS-FD) as basis functions, combined with modified method of characteristics. These techniques are combined with careful design for the spatial derivative operators in the momentum flux equation, according to a general criterion for the exact preservation of the “lake at rest” solution in general mesh-based and meshless numerical schemes for the strong form of the shallow-water equations with bottom topography. Both structured and unsructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports and maximal permissible degree of the polynomials in RBFPHS-FD.

A TVD discretization method for shallow water equations: Numerical simulations of tailing dam break

In view of the disastrous consequences of tailings dam break and its unique evolutionary process in complex areas, this paper constructs two-dimensional shallow water equations, rheological equations and mathematical models of tailings sand flows on the basis of Navier–Stokes equations (N–S equations). It performs total variation diminishing (TVD) discretization on these equations, develops forward simulation programs in MATLAB2016 and conducts numerical analyses on three kinds of dam breaks (ideal dam break, asymmetric dam break and dam break with obstacles in the downstream area). The results show that TVD discretization is effective in capturing shock waves. According to the analysis on consequences of Huangmailing Tailings Dam break, the author obtains the maximum distance of tailings sand flow, the flow rate of tailings and the time that tailings reach destinations in the downstream area, thereby providing scientific basis for disaster analyses on similar tailings dam breaks and supplying technical support for emergency rescues after disasters.

Dynamically adaptive tree grid modeling for simulation and visualization of rainwater overland flow

SummaryModeling and visualization of a rainwater overland flow is an important tool for a risk assessment, preparation, evacuation planning, and real‐time forecasting of flood warning. The objective of this research is to develop a numerical software to visualize the rainwater overland flow based on a finite volume method for shallow water equations and in combination with the dynamically adaptive tree grid technique and the dynamic domain‐defining method. The obtained simulations for several experiments were tested and compared with results in literature, both theoretical and experimental results. The comparisons with non‐adaptive grids show that the developed algorithm for simulation is very efficient and has a potential for practical usages, in terms of computational times and accuracy. Copyright © 2015 John Wiley & Sons, Ltd.

Multi-level Monte Carlo finite volume method for shallow water equations with uncertain parameters applied to landslides-generated tsunamis

Two layer Savage–Hutter type shallow water PDEs model flows such as tsunamis generated by rockslides. On account of heterogeneities in the composition of the granular matter, these models contain uncertain parameters like the ratio of densities of layers, Coulomb and interlayer friction. These parameters are modeled statistically and quantifying the resulting solution uncertainty (UQ) is a crucial task in geophysics. We propose a novel paradigm for UQ that combines the recently developed IFCP spatial discretizations with the recently developed Multi-level Monte Carlo (MLMC) statistical sampling method and provides a fast, accurate and computationally efficient framework to compute statistical quantities of interest. Numerical experiments, including realistic simulations of the Lituya Bay mega tsunami, are presented to illustrate the robustness of the proposed UQ algorithm.

Recent advances on the discontinuous Galerkin method for shallow water equations with topography source terms

We consider in this work the discontinuous Galerkin discretization of the nonlinear shallow water equations on unstructured triangulations. In the recent years, several improvements have been made in the quality of the discontinuous Galerkin approximations for the shallow water equations. In this paper, we first perform a review of the recent methods introduced to ensure the preservation of motionless steady states and robust computations. We then suggest an efficient combination of ingredients that leads to a simple high-order robust and well-balanced scheme, based on the alternative formulation of the equations known as the pre-balanced shallow water equations. We show that the preservation of the motionless steady states can be achieved, for an arbitrary order of polynomial expansion. Additionally, the preservation of the positivity of the water height is ensured using the recent method introduced in (Zhang et al., 2012). An extensive set of numerical validations is performed to highlight the efficiency of these approaches. Some accuracy, CPU-time and convergence studies are performed, based on comparisons with analytical solutions or validations against experimental data, for several test cases involving steady states and occurrence of dry areas. Some comparisons with a recent Finite-Volume MUSCL approach are also performed.

Quadtree flood simulations with sub-grid digital elevation models

Flooding is an increasing hazard to society and good governance now implies careful water management in terms of design, planning and control of urban and rural areas. This requires that rainstorms, extreme water levels and so on are taken into account with relevant precision. A great aid is the existence of graphical information systems with raster-based digital elevation models (DEMs). Modern technology such as Lidar means that DEMs are of ever-increasing resolution. This paper describes how, without adaptations, a DEM can be used efficiently for detailed 2D flooding simulations. The method is based on four components: the sub-grid method; bottom friction derived from the divided channel method; the finite-volume staggered grid method for shallow water equations with rapidly varying flows; and quadtrees.

A parallel well-balanced finite volume method for shallow water equations with topography on the cubed-sphere

A finite volume scheme for the global shallow water model on the cubed-sphere mesh is proposed and studied in this paper. The new cell-centered scheme is based on Osher’s Riemann solver together with a high-order spatial reconstruction. On each patch interface of the cubed-sphere only one layer of ghost cells is needed in the scheme and the numerical flux is calculated symmetrically across the interface to ensure the numerical conservation of total mass. The discretization of the topographic term in the equation is properly modified in a well-balanced manner to suppress spurious oscillations when the bottom topography is non-smooth. Numerical results for several test cases including a steady-state nonlinear geostrophic flow and a zonal flow over an isolated mountain are provided to show the flexibility of the scheme. Some parallel implementation details as well as some performance results on a parallel supercomputer with more than one thousand processor cores are also provided.

Hybrid finite element/volume method for shallow water equations

AbstractA hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two‐fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non‐conservation form and solve the non‐linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell‐center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix‐free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method. Copyright © 2010 John Wiley & Sons, Ltd.