Shallow Water Equation Solver Research Articles

An approximate Riemann solver for shallow water equations and heat advection in horizontal centrifugal casting

An approximate Riemann solver was developed for solving modified shallow water equations (SWE) and energy transport describing the average flow dynamics of a single layer spreading inside a horizontally rotating cylinder. The numerical model was particularly developed for simulating the horizontal centrifugal casting (HSC) of the outer shell of a work roll. The SWE were derived in the rotating frame of reference; therefore, fictitious forces (the centrifugal force and the Coriolis force) were considered. In addition, other forces such as the bed shear force, the force of gravity, the wind shear force and forces arising from the variable liquid/solid interface were taken into account. The Jacobian matrix of the nonlinear hyperbolic system of PDEs was decomposed into a set of eigenvalues and corresponding eigenvectors using standard and corrected Roe averages. A Harten–Hyman entropy fix was used to prevent expansion shocks (entropy violating solutions) typically occurring during transonic rarefactions. Source terms were applied as a stationary discontinuity and were physically bounded and well-balanced for steady states (producing non-oscillatory solutions). Each wave was upwinded using the explicit Godunov’s method. The high resolution corrections with flux limiters were used to achieve second order of accuracy and dispersion free solutions at discontinuities. In addition to the Riemann solver, a central scheme FV model was used to solve the heat diffusion inside the cylinder (mold) and partially solidified liquid layer, coupled with the solidification model. Several simulations were performed, results were analyzed and discussed.

The SIPSim implicit parallelism model and the SkelGIS library

SummaryScientific simulations give rise to complex codes where data size and computation time become very important issues, and sometimes a scientific barrier. Thus, parallelization of scientific simulations becomes a significant work. Many time and human efforts are deployed to produce efficient parallel programs. But still, many simulations could not be parallelized because of lack of time to learn parallel programming or lack of human resources. Therefore, aiding parallelization through abstracted parallelism or implicit parallelism has become a main topic in computer science. Many implicit parallelism solutions have been proposed such as algorithmic skeletons libraries, domain‐specific languages or specific libraries. In this paper is introduced a new type of solution to give a totally transparent access to parallel programming for non‐computer scientists of the domain of numerical simulations. This solution is an implicit parallelism model, called Structured Implicit Parallelism on scientific Simulations (SIPSim). After a description of the SIPSim model, this paper presents the implementation of the model, as a C++ templated library called SkelGIS, for two different cases of simulations: simulations on Cartesian meshes and simulations of two physical phenomena linked through a network. For each case, the implementation of the SIPSim components are described, and a simple simulation example is given. SkelGIS is then evaluated on two real cases, one for each case, first on the resolution of shallow water equations and second on an arterial blood flow simulation. To clearly state on SkelGIS performance and its ease of programming, different experiments on both cases are evaluated. Copyright © 2015 John Wiley & Sons, Ltd.

A Cell-Integrated Semi-Lagrangian Semi-Implicit Shallow-Water Model (CSLAM-SW) with Conservative and Consistent Transport

Abstract A Cartesian semi-implicit solver using the Conservative Semi-Lagrangian Multitracer (CSLAM) transport scheme is constructed and tested for shallow-water (SW) flows. The SW equations solver (CSLAM-SW) uses a discrete semi-implicit continuity equation specifically designed to ensure a conservative and consistent transport of constituents by avoiding the use of a constant mean reference state. The algorithm is constructed to be similar to typical conservative semi-Lagrangian semi-implicit schemes, requiring at each time step a single linear Helmholtz equation solution and a single application of CSLAM. The accuracy and stability of the solver is tested using four test cases for a radially propagating gravity wave and two barotropically unstable jets. In a consistency test using the new solver, the specific concentration constancy is preserved up to machine roundoff, whereas a typical formulation can have errors many orders of magnitude larger. In addition to mass conservation and consistency, CSLAM-SW also ensures shape preservation by combining the new scheme with existing shape-preserving filters. With promising SW test results, CSLAM-SW shows potential for extension to a nonhydrostatic, fully compressible system solver for numerical weather prediction and climate models.

Influence of Total-Variation-Diminishing Slope Limiting on Local Discontinuous Galerkin Solutions of the Shallow Water Equations

Finite volume (FV) slope limiting is essential to stabilize discontinuous Galerkin (DG) solutions despite a number of side effects such as local loss of accuracy and increased run-time cost. These side effects have been experienced with DG solutions to the homogeneous system of conservation laws and are usually accepted as long as they do not affect the reliability of the numerical predictions and provide a better stability property. They have also led to the concept of localizing the slope-limiting process. When a model is applied to simulate the flow problems that necessitate the solution of the inhomogeneous shallow water equations (SWEs) with/without flooding and drying, the slope-limiting routine can have extra adverse effects on the local DG framework. These effects, causes, and implications on DG solutions to the SWEs have not yet received adequate attention, and a full investigation is therefore needed. With the aim of improving a second-order Runge-Kutta discontinuous Galerkin (RKDG2) SWE solver, the effects of FV slope limiting on the RKDG2 flow solutions were analyzed. Three versions of the RKDG2 scheme were configured: the first and second versions were respectively associated with the local limiting (LL) and global limiting (GL) processes, which both use limiter-free linear front tracking (LFT) to trace a wet/dry front, and the third was implemented with the local limiting process but with constant front-tracking (CFT) of the wet/dry front. Selected hypothetical and practical hydraulic tests were investigated to demonstrate the effects of different limiting processes and front-tracking techniques.

Coupled wave action and shallow-water modelling for random wave runup on a slope

Wave runup statistics on beaches, dunes and coastal structures are needed for coastal management and engineering designs. Spectral wave energy–action models are widely used to predict wave propagation in deep and shallow water, incorporating the effects of refraction, bed friction, breaking and wave–wave interactions. To calculate nearshore runup or overtopping, wave-by-wave or phase-resolving modelling is necessary as spectral, phase-averaged models only give the spectral evolution and wave-induced setup. Herein, unidirectional wave propagation up a uniform slope predicted by the Simulating WAves Nearshore (SWAN) model is compared with experimental data. The surface elevation time series near the breaking position is then created from the predicted nearshore spectra and input to a non-linear shallow-water equation solver to give statistical runup predictions which are also compared with experimental data. Sensitivity to the offshore position for wave input is investigated. Runup statistics are shown to be reasonably well predicted.

수리실험 및 수치모의를 이용한 제방붕괴 흐름해석

Abrupt and gradual levee breach analyses on the flat domain were implemented by laboratory experiments and numerical simulations. To avoid the reflective wave from the side wall the experiment was performed in a large domain surrounded by waterway. A numerical model was developed for solving the two-dimensional gradual levee breach flow. The results of the numerical simulation developed in this study showed good agreement with those of the experimental data. However, even if the numerical schemes effectively replicated the trends of the observed water depth for the first shock, there were little differences for the second shock. In addition, even though the model considered the Smagorinsky horizontal eddy viscosity, the location and height of the hydraulic jump in the numerical simulation were not fairly well agree with experimental measurements. This shows the shallow water equation solver has a limitation which does not exactly reproduce the energy dissipation from the hydraulic jump. Further study might be required, considering the energy dissipation due to the hydraulic jump or transition flow from reflective wave.

A simplified adaptive Cartesian grid system for solving the 2D shallow water equations

SUMMARYThis paper presents a new simplified grid system that provides local refinement and dynamic adaptation for solving the 2D shallow water equations (SWEs). Local refinement is realized by simply specifying different subdivision levels to the cells on a background uniform coarse grid that covers the computational domain. On such a non‐uniform grid, the structured property of a regular Cartesian mesh is maintained and neighbor information is determined by simple algebraic relationships, i.e. data structure becomes unnecessary. Dynamic grid adaptation is achieved by changing the subdivision level of a background cell. Therefore, grid generation and adaptation is greatly simplified and straightforward to implement. The new adaptive grid‐based SWE solver is tested by applying it to simulate three idealized test cases and promising results are obtained. The new grid system offers a simplified alternative to the existing approaches for providing adaptive mesh refinement in computational fluid dynamics. Copyright © 2011 John Wiley & Sons, Ltd.

Parallel hyperbolic PDE simulation on clusters: Cell versus GPU

Increasingly, high-performance computing is looking towards data-parallel computational devices to enhance computational performance. Two technologies that have received significant attention are IBM's Cell Processor and NVIDIA's CUDA programming model for graphics processing unit (GPU) computing. In this paper we investigate the acceleration of parallel hyperbolic partial differential equation simulation on structured grids with explicit time integration on clusters with Cell and GPU backends. The message passing interface (MPI) is used for communication between nodes at the coarsest level of parallelism. Optimizations of the simulation code at the several finer levels of parallelism that the data-parallel devices provide are described in terms of data layout, data flow and data-parallel instructions. Optimized Cell and GPU performance are compared with reference code performance on a single x86 central processing unit (CPU) core in single and double precision. We further compare the CPU, Cell and GPU platforms on a chip-to-chip basis, and compare performance on single cluster nodes with two CPUs, two Cell processors or two GPUs in a shared memory configuration (without MPI). We finally compare performance on clusters with 32 CPUs, 32 Cell processors, and 32 GPUs using MPI. Our GPU cluster results use NVIDIA Tesla GPUs with GT200 architecture, but some preliminary results on recently introduced NVIDIA GPUs with the next-generation Fermi architecture are also included. This paper provides computational scientists and engineers who are considering porting their codes to accelerator environments with insight into how structured grid based explicit algorithms can be optimized for clusters with Cell and GPU accelerators. It also provides insight into the speed-up that may be gained on current and future accelerator architectures for this class of applications. Program summary Program title: SWsolver Catalogue identifier: AEGY_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGY_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GPL v3 No. of lines in distributed program, including test data, etc.: 59 168 No. of bytes in distributed program, including test data, etc.: 453 409 Distribution format: tar.gz Programming language: C, CUDA Computer: Parallel Computing Clusters. Individual compute nodes may consist of x86 CPU, Cell processor, or x86 CPU with attached NVIDIA GPU accelerator. Operating system: Linux Has the code been vectorised or parallelized?: Yes. Tested on 1-128 x86 CPU cores, 1-32 Cell Processors, and 1-32 NVIDIA GPUs. RAM: Tested on Problems requiring up to 4 GB per compute node. Classification: 12 External routines: MPI, CUDA, IBM Cell SDK Nature of problem: MPI-parallel simulation of Shallow Water equations using high-resolution 2D hyperbolic equation solver on regular Cartesian grids for x86 CPU, Cell Processor, and NVIDIA GPU using CUDA. Solution method: SWsolver provides 3 implementations of a high-resolution 2D Shallow Water equation solver on regular Cartesian grids, for CPU, Cell Processor, and NVIDIA GPU. Each implementation uses MPI to divide work across a parallel computing cluster. Additional comments: Sub-program numdiff is used for the test run.

Sensitivity Analysis for Hydraulic Models

Sensitivity analysis is well recognized as being an important aspect of the responsible use of hydraulic models. This paper reviews a range of methods for sensitivity analysis. Two applications, one to a simple pipe bend example and the second to an advanced Shallow Water Equation solver, illustrate the deficiencies of standardized regression coefficients in the context of functionally nonlinear models. Derivatives and other local methods of sensitivity analysis are shown to give an incomplete picture of model response over the range of variability in the model inputs. The use of global variance-based sensitivity analysis is shown to be more general in its applicability and in its capacity to reflect nonlinear processes and the effects of interactions among variables.

Flood Inundation Modeling with an Adaptive Quadtree Grid Shallow Water Equation Solver

Flood risk studies require hydraulic modeling in order to estimate flow depths and other hydraulic variables in the floodplain for a wide range of input conditions. Currently there is a need to improve the computational efficiency of fully two-dimensional numerical models for large-scale flood simulation. This paper describes an adaptive quadtree grid-based shallow water equation solver and demonstrates its capability for flood inundation modeling. Due to the grid dynamically adapting to dominant flow features such as steep water surface gradients and wet-dry fronts, the approach is both efficient and accurate. The quadtree model is applied to a realistic scenario of flood inundation over an urban area of 36km2, resulting from the flood defenses breaching at Thamesmead on the River Thames, United Kingdom. The results of the simulation are in close agreement with alternative predictions obtained using the commercially available software TUFLOW.

Modelling the formation and the long‐term behavior of rip channel systems from the deformation of a longshore bar

A nonlinear numerical model based on a wave‐ and depth‐averaged shallow water equation solver with wave driver, sediment transport, and bed updating is used to investigate the long‐term evolution of rip channel systems appearing from the deformation of a longshore bar. Linear and nonlinear regimes in the morphological evolution have been studied. In the linear regime, a crescentic bar system emerges as a free instability. In the nonlinear regime, merging/splitting in bars and saturation of the growth are obtained. In spite of excluding undertow and wave‐asymmetry sediment transport, the initial crescentic bar system reorganizes to form a large‐scale and shore‐attached transverse or oblique bar system, which is found to be a dynamical equilibrium state of the beach system. Thus the basic morphological transitions “Longshore Bar and Trough” → “Rhythmic Bar and Beach” → “Transverse Bar and Rip” described by earlier conceptual models are here reproduced. The study of the physical mechanisms allows us to understand the role of the different transport modes: The advective part induces the formation of crescentic bars and megacusps, and the bedslope transport damps the instability. Both terms contribute to the attachment of the megacusps to the crescentic bars. Depending on the wave forcing, the bar wavelength ranges between 180 and 250 m (165 and 320 m) in the linear (nonlinear) regime.

Numerical viscosity reduction in the resolution of the shallow water equations with turbulent term

AbstractA first‐order finite volume model for the resolution of the 2D shallow water equations with turbulent term is presented. An upwind discretization of the equations that include the turbulent term is carried out. A method to reduce the excess of numerical viscosity (or diffusion) produced by the upwinding of the flux term is proposed. Two different discretizations of the turbulent term are compared, and results for uniform distributions of the viscosity are presented. Finally, two discretizations of the time derivative which are more efficient than Euler's are proposed and compared. Copyright © 2008 John Wiley & Sons, Ltd.

Generation and nonlinear evolution of shore-oblique/transverse sand bars

The coupling between topography, waves and currents in the surf zone may self-organize to produce the formation of shore-transverse or shore-oblique sand bars on an otherwise alongshore uniform beach. In the absence of shore-parallel bars, this has been shown by previous studies of linear stability analysis, but is now extended to the finite-amplitude regime. To this end, a nonlinear model coupling wave transformation and breaking, a shallow-water equations solver, sediment transport and bed updating is developed. The sediment flux consists of a stirring factor multiplied by the depth-averaged current plus a downslope correction. It is found that the cross-shore profile of the ratio of stirring factor to water depth together with the wave incidence angle primarily determine the shape and the type of bars, either transverse or oblique to the shore. In the latter case, they can open an acute angle against the current (up-current oriented) or with the current (down-current oriented). At the initial stages of development, both the intensity of the instability which is responsible for the formation of the bars and the damping due to downslope transport grow at a similar rate with bar amplitude, the former being somewhat stronger. As bars keep on growing, their finite-amplitude shape either enhances downslope transport or weakens the instability mechanism so that an equilibrium between both opposing tendencies occurs, leading to a final saturated amplitude. The overall shape of the saturated bars in plan view is similar to that of the small-amplitude ones. However, the final spacings may be up to a factor of 2 larger and final celerities can also be about a factor of 2 smaller or larger. In the case of alongshore migrating bars, the asymmetry of the longshore sections, the lee being steeper than the stoss, is well reproduced. Complex dynamics with merging and splitting of individual bars sometimes occur. Finally, in the case of shore-normal incidence the rip currents in the troughs between the bars are jet-like while the onshore return flow is wider and weaker as is observed in nature.

Particle mixing and reactive front motion in unsteady open shallow flow – Modelled using singular value decomposition

Passive and active tracers are used to examine particle mixing and reactive front dynamics in an open shallow flow of water past a circular cylinder. A quadtree grid based Godunov-type shallow water equation solver predicts the unsteady flow hydrodynamics of the wake behind the cylinder. The resulting periodic flow field consisting of a von Kármán vortex street is decomposed and stored over one oscillatory period using Singular Value Decomposition (SVD). Particles are advected according to the reconstructed flow field from the SVD modes, with continuous spatial velocity information obtained via bilinear interpolation. Passive particle dynamics driven by different SVD flow modes is investigated, and it is found that the flow field recovered from the mean flow and the first pair of time varying modes is adequate to represent the complicated dynamical properties induced by the original flow field. Active autocatalytic reaction, A + B → 2 B, is incorporated into the particle advection model, assuming surface reaction. Active particles are found to trace out an expanded version of the unstable manifold of the chaotic saddle in the wake, in qualitative agreement with published analytical results. The numerical model is applicable to mixing and transport processes in more complicated shallow environmental flows.

Chaotic mixing in a basin due to a sinusoidal wind field

AbstractThis paper presents a novel numerical study of wind‐induced mixing in a circular shallow lake. The wind‐induced hydrodynamic field is first predicted by a Godunov‐type shallow water equation solver based on a quadtree grid. Then the horizontal mixing processes are analysed in terms of chaotic advection. The particle dynamics under a sinusoidal wind field are found to change from regular to chaotic states with increasing period of wind oscillation. The numerical approach is applicable to a wide range of environmental flows. Copyright © 2005 John Wiley & Sons, Ltd.

An entropy-correction free solver for non-homogeneous shallow water equations

In this work we introduce an accurate solver for the Shallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.

A flux‐splitting solver for shallow water equations with source terms

AbstractThis paper introduces a stable flux‐splitting solver for one‐dimensional (1D) shallow water equations. This solver is specifically designed to satisfy a strengthened consistency condition for stationary solutions that ensures the stability and accuracy of the scheme. It applies to channels with variable depth and width, including terms modelling friction at bottom and vertical walls. Some numerical tests by comparison to both analytical solutions and experimental measurements show the good performances of the scheme. Copyright © 2003 John Wiley & Sons, Ltd.