System Of Shallow Water Equations Research Articles

Physics‐Informed Neural Networks for the Augmented System of Shallow Water Equations With Topography

AbstractPhysics‐informed neural networks (PINNs) are gaining attention as an alternative approach to solve scientific problems governed by differential equations. This work aims at assessing the effectiveness of PINNs to solve a set of partial differential equations for which this method has never been considered, namely the augmented shallow water equations (SWEs) with topography. Differently from traditional SWEs, the bed elevation is considered as an additional conserved variable, and therefore one more equation expressing the fixed‐bed condition is included in the system. This approach allows the PINN model to leverage automatic differentiation to compute the bed slopes by learning the topographical information during training. PINNs are here tested for different one‐dimensional cases with non‐flat topography, and results are compared with analytical solutions. Though some limitations can be highlighted, PINNs show a good accuracy for the depth and velocity predictions even in the presence of non‐horizontal bottom. The solution of the augmented system of SWEs can therefore be regarded as a suitable alternative strategy to deal with flows over complex topography using PINNs, also in view of future extensions to realistic problems.

A one-dimensional augmented Shallow Water Equations system for channels of arbitrary cross-section

This work provides a new formulation of the one-dimensional augmented Shallow Water Equations system for open channels and rivers with arbitrarily shaped cross sections, suitable for numerical integration when discontinuous geometry is encountered. The additional variable considered can be the bottom elevation, a reference width, a shape coefficient, or a vector containing these or other geometric parameters. The appropriate numerical method, which is well suited to coupling with the mathematical one, is a path-conservative method, capable of reconstructing the behaviour of physical and geometrical variables at the cell boundaries, where the discrete solution of hyperbolic systems of equations is discontinuous. A nonlinear path suitable for the shallow water context is adopted. The resulting model is shown to be well-balanced and accurate to the second order and is further validated against analytical solutions related to channels with power-law cross-sections, specifically for dam break patterns over a variable-width channel and the run-up dynamics of long water waves over sloping bays.

On the data assimilation of initial distribution for 2-dimensionalshallow-water equation model

Abstract The utilization of data assimilation (DA) techniques is prevalent in marine meteorology for the purpose of estimating the complete state of the system. This is done to address the practical limitations associated with measurement data, which can only be specified at finite number of discrete points within a limited domain. We develop an efficient DA algorithm to reconstruct the initial state of the shallow-water equations (SWE) within a 2-dimensional rectangular domain using sparse spatial measurement data. Our algorithm takes into account both the complete Coriolis force and the ocean bottom topography in the SWE model, resulting in accurate recovery of the initial status. After establishing the uniqueness of the solution to the nonlinear SWE with appropriate boundary conditions, we proceed to establish the conservation laws for the suitably defined energy quantity for this traveling wave system. This generalization of the known conservation laws for the simplified SWE system, which ignores the Coriolis force and topography, allows us to reveal the influence of nonconstant sea floor topography on wave propagation. In order to restore the initial state through the minimization of a cost functional using DA techniques, we proceed by deriving the adjoint problem for our iteration process. Additionally, we establish a discrete scheme for the governing equations in the Arakawa C-grid framework, from which we rigorously derive the error associated with energy conservation in discrete form. The numerical implementations are also provided to validate our proposed scheme through the verification of energy conservation and the reconstruction effect of the initial state for various configurations.

A novel method to study time fractional coupled systems of shallow water equations arising in ocean engineering

<abstract><p>This study investigates solutions for the time-fractional coupled system of the shallow-water equations. The shallow-water equations are employed for the purpose of elucidating the dynamics of water motion in oceanic or sea environments. Also, the aforementioned system characterizes a thin fluid layer that maintains a hydrostatic equilibrium while exhibiting uniform density. Shallow water flows have a vertical dimension that is considerably smaller in magnitude than the typical horizontal dimension. In the current work, we employ an innovative and effective technique, known as the natural transform decomposition method, to obtain the solutions for these fractional systems. The present methodology entails the utilization of both singular and non-singular kernels for the purpose of handling fractional derivatives. The Banach fixed point theorem is employed to demonstrate the uniqueness and convergence of the obtained solution. The outcomes obtained from the application of the suggested methodology are compared to the exact solution and the results of other numerical methods found in the literature, including the modified homotopy analysis transform method, the residual power series method and the new iterative method. The results obtained from the proposed methodology are presented through the use of tabular and graphical simulations. The current framework effectively captures the behavior exhibited by different fractional orders. The findings illustrate the efficacy of the proposed method.</p></abstract>

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.

Avaliação de formulações de taxa de deformação através de escoamentos permanentes e uniformes de fluidos não-Newtonianos no contexto de equações de águas rasas

ABSTRACT Non-Newtonian rheology effects, such as pseudoplasticity and viscoplasticity, are understood as shear stresses, incorporated to the energy slope term in the Shallow-Water Equations (SWE). However, non-Newtonian shear stresses are dependent of the shear rate, whose formulation is a function of the gradient of the velocity profile in the bottom. This study investigated two shear rate formulations that are commonly applied in the SWE literature: 1) a non-parameterized function; and 2) a function based on the Herschel-Bulkley rheological model. Their influence in steady uniform flows of non-Newtonian fluids was evaluated through numerical-theoretical comparisons. A Lax-Friedrichs scheme was implemented to solve the SWE system and allowed employing the shear rate formulations. Experimental tests were carried out and numerical simulations of hypothetical scenarios were performed. It was found that the non-parameterized formulation presented deviation in normal depth up to 14% in comparison with theoretical solution, while the formulation based on the Herschel-Bulkley model provided a good agreement, corroborated by punctual Computational Fluid Dynamics simulations (deviation less than 2%) and experimental data. The ratio of both shear rate formulations is strongly correlated to the deviation of normal depth, indicating that the non-parameterized shear rate function does not provide an acceptable result in the steady uniform flow.

Existence of Global Classical Solutions for the Saint-Venant Equations

Nowadays, investigations of the existence of global classical solutions for non linear evolution equations is a topic of active mathematical research. In this article, we are concerned with a classical system of shallow water equations which describes long surface waves in a fluid of variable depth. This system was proposed in 1871 by Adh\'{e}mar Jean-Claude Barr\'{e} de Saint-Venant. Namely, we investigate an initial value problem for the one dimensional Saint-Venant equations. We are especially interested in question of what sufficient conditions the initial data and the topography of the bottom must verify in~order that the considered system has global classical solutions. In order to prove our main results we use a~new topological approach based on the fixed point abstract theory of the sum of two operators in Banach spaces. This basic and new idea yields global existence theorems for many of the interesting equations of mathematical physics.

Well-balanced positivity preserving adaptive moving mesh central-upwind schemes for the Saint-Venant system

We extend the adaptive moving mesh (AMM) central-upwind schemes recently proposed in Kurganov et al. [Commun. Appl. Math. Comput. 3 (2021) 445–479] in the context of one- (1-D) and two-dimensional (2-D) Euler equations of gas dynamics and granular hydrodynamics, to the 1-D and 2-D Saint-Venant system of shallow water equations. When the bottom topography is nonflat, these equations form hyperbolic systems of balance laws, for which a good numerical method should be capable of preserving a delicate balance between the flux and source terms as well as preserving the nonnegativity of water depth even in the presence of dry or almost dry regions. Therefore, in order to extend the AMM central-upwind schemes to the Saint-Venant systems, we develop special positivity preserving reconstruction and evolution steps of the AMM algorithms as well as special corrections of the solution projection step in (almost) dry areas. At the same time, we enforce the moving mesh to be structured even in the case of complicated 2-D computational domains. We test the designed method on a number of 1-D and 2-D examples that demonstrate robustness and high resolution of the proposed numerical approach.

Some quantitative characteristics of error covariance for Kalman filters

Some quantitative characteristics of error covariance are studied for linear Kalman filters. These quantitative characteristics include the peak value and location in the matrix, the decay rate from peak to bottom, and some algebraic constraints of the elements in the covariance matrix. We mathematically prove a matrix upper bound and its quantitative characteristics for the error covariance of Kalman filters. Computational methods are developed to numerically estimate the elements in a matrix upper bound and its decay rate. The quantitative characteristics and the computational methods are illustrated using three examples, two linear systems and one nonlinear system of shallow water equations.

Chislennoye modelirovaniye shtormovogo nagona 15 noyabrya 2019 goda na yuge ostrova Sakhalin

Purpose . Investigation of the storm surge in Korsakov in the southern part of the Sakhalin Island on November 15, 2019 and comparison of the results of its numerical simulation with the data of in situ measurements constitute the aim of the article. Methods and Results . In situ measurements of the storm surge in Korsakov (the Sakhalin region) were performed and the data on the flooded area dimensions were collected. A storm period on the Sakhalin Island is almost the annual event in an autumn-winter season. The severe storm that happened in the southern Sakhalin region on November 15, 2019 led to flooding of the port territory in Korsakov. Due to the NAMI-DANCE computational complex, the storm surge was numerically simulated within the framework of the system of shallow water equations in the spherical coordinates on the rotating Earth with the regard for the friction force and the atmospheric effect. The calculations included the data on temporal and spatial distribution of the wind speed at the altitude 10 m taken from the Climate Forecast System Reanalysis database. The data on the atmospheric pressure were not applied in simulations since the atmosphere pressure gradient at the area under study was small. The simulation was carried out in the course of three days. The simulations showed that in 20 hours after the wind forcing had started, the water level in the port increased up to its maximum values, and did not fall the whole day. The water level maximum heights were concentrated in the southwestern part of the Aniva Bay. At that the calculated current speeds reached 2 m/s. During the storm, at the wind speed up to 15 m/s, the storm surge height in the Korsakov port area constituted 1.7 m, whereas the width of the flooded zone was up to 200 m. These results are confirmed well by the in situ measurement data. Conclusions . The simulation values of the power characteristics for the above-mentioned storm are represented in the paper. The Froude number square reaches 0.03 in the Korsakov city port area, and spatial distribution of the wave strength moment is up to 1 m3/s2. Field measurements and eyewitness reports confirm the evidence of a powerful impact of a storm surge upon the port constructions.

Numerical Simulation of the Storm Surge at the Sakhalin Island Southern Part on November 15, 2019

Purpose. Investigation of the storm surge in Korsakov in the southern part of the Sakhalin Island on November 15, 2019 and comparison of the results of its numerical simulation with the data of in situ measurements constitute the aim of the article. Methods and Results. In situ measurements of the storm surge in Korsakov (the Sakhalin region) were performed and the data on the flooded area dimensions were collected. A storm period on the Sakhalin Island is almost the annual event in an autumn-winter season. The severe storm that happened in the southern Sakhalin region on November 15, 2019 led to flooding of the port territory in Korsakov. Due to the NAMI-DANCE computational complex, the storm surge was numerically simulated within the framework of the system of shallow water equations in the spherical coordinates on the rotating Earth with the regard for the friction force and the atmospheric effect. The calculations included the data on temporal and spatial distribution of the wind speed at the altitude 10 m taken from the Climate Forecast System Reanalysis database. The data on the atmospheric pressure were not applied in simulations since the atmosphere pressure gradient at the area under study was small. The simulation was carried out in the course of three days. The simulations showed that in 20 hours after the wind forcing had started, the water level in the port increased up to its maximum values, and did not fall the whole day. The water level maximum heights were concentrated in the southwestern part of the Aniva Bay. At that the calculated current speeds reached 2 m/s. During the storm, at the wind speed up to 15 m/s, the storm surge height in the Korsakov port area constituted 1.7 m, whereas the width of the flooded zone was up to 200 m. These results are confirmed well by the in situ measurement data. Conclusions. The simulation values of the power characteristics for the above-mentioned storm are represented in the paper. The Froude number square reaches 0.03 in the Korsakov city port area, and spatial distribution of the wave strength moment is up to 1 m3/s2. Field measurements and eyewitness reports confirm the evidence of a powerful impact of a storm surge upon the port constructions.

Local tsunami run-up depending on initial localization of the landslide body at submarine slope

The numerical simulation of tsunami induced by layer-by-layer sliding of submarine slope with various initial location of landslide body is performed. For the landslide body, an elastoplastic model with layered sediments taking into account the porosity and deconsolidation of the landslide mass, resting on a relatively rigid base, is used. The characteristic parameters of the above model correspond to those of the landslide and tsunami event in the Corinth Bay on February 7, 1963 modeled by the authors (Papadopoulos et al. 2007). To numerically represent the dynamics of landslide motion numerical code FLAC is used, which in contrast to the finite element method implements an explicit finite-difference scheme for solving three-dimensional problems of continuum mechanics, and allows simulation of the nonlinear behavior of porous-saturated grounds under conditions of plastic flow above the yield stress. Because of plane slope conditions, the nonlinear system of shallow water equations is used for the numerical simulation of tsunami. The results demonstrate that at each time moment, the tsunami runup occurs at novel surface of the coastal slope that leads to complex repositioning of the shoreline point that depends on initial location of the landslide volume. Such an observation is absent in conventional (e.g. rigid block, viscoplastic etc.) models The results of the work demonstrate a rather physical picture of the process under consideration. More importantly modeling results of this work may help in identifying the generating mechanism of historical or future landslide tsunamis.

Zeroth‐order conservation laws of two‐dimensional shallow water equations with variable bottom topography

AbstractWe classify zeroth‐order conservation laws of systems from the class of two‐dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth‐order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating the classification of zeroth‐order conservation laws of these systems to the classification of their Lie symmetries. We also construct generating sets of such conservation laws under action of Lie symmetries.

One-dimensional model of turbulent flow of non-Newtonian drilling mud in non-prismatic channels

One-dimensional model of non-Newtonian turbulent flow in a non-prismatic channel is challenging due to the difficulty of accurately accounting for flow properties in the 1-D model. In this study, we model the 1-D Saint–Venant system of shallow water equations for water-based drilling mud (non-Newtonian) in open Venturi channels for steady and transient conditions. Numerically, the friction force acting on a fluid in a control volume can be subdivided, in the 1-D drilling mud modelling and shallow water equations, into two terms: external friction and internal friction. External friction is due to the wall boundary effect. Internal friction is due to the non-Newtonian viscous effect. The internal friction term can be modelled using pure non-Newtonian viscosity models, and the external friction term using Newtonian wall friction models. Experiments were carried out using a water-based drilling fluid in an open Venturi channel. Density, viscosity, flow depth, and flow rate were experimentally measured. The developed approach used to solve the 1-D non-Newtonian turbulence model in this study can be used for flow estimation in oil well return flow.

Lie Symmetries, One-Dimensional Optimal System and Group Invariant Solutions for the Ripa System

Abstract A complete symmetry group classification for the system of shallow water equations with the horizontal temperature gradient, also known as Ripa system, is presented. A rigorous and systematic procedure based on the general invariants of the adjoint representation is used to construct the one-dimensional optimal system of the Lie algebra. The complete inequivalence class of the group invariant solutions are obtained by using the one-dimensional optimal system. One such solution of the Ripa system is used to study the evolutionary behaviour of the discontinuity wave.

An asymptotic preserving scheme for the two-dimensional shallow water equations with Coriolis forces

We consider the two-dimensional Saint-Venant system of shallow water equations with Coriolis forces. We focus on the case of a low Froude number, in which the system is stiff and conventional explicit numerical methods are extremely inefficient and often impractical. Our goal is to design an asymptotic preserving (AP) scheme, which is uniformly asymptotically consistent and stable for a broad range of (low) Froude numbers. The goal is achieved using the flux splitting proposed in [Haack et al., Commun. Comput. Phys., 12 (2012), pp. 955–980] in the context of isentropic Euler and Navier-Stokes equations. We split the flux into the stiff and nonstiff parts and then use an implicit-explicit approach: apply an explicit hyperbolic solver (we use the second-order central-upwind scheme) to the nonstiff part of the system while treating the stiff part of it implicitly. Moreover, the stiff part of the flux is linear and therefore we reduce the implicit stage of the proposed method to solving a Poisson-type elliptic equation, which is discretized using a standard second-order central difference scheme.We conduct a series of numerical experiments, which demonstrate that the developed AP scheme achieves the theoretical second-order rate of convergence and the time-step stability restriction is independent of the Froude number. This makes the proposed AP scheme an efficient and robust alternative to fully explicit numerical methods.

A New Approach for Designing Moving-Water Equilibria Preserving Schemes for the Shallow Water Equations

We construct a new second-order moving-water equilibria preserving central-upwind scheme for the one-dimensional Saint-Venant system of shallow water equations. The idea is based on a reformulation of the source terms as integral in the flux function. Reconstruction of the flux variable yields then a third order equation that can be solved exactly. This procedure does not require any further modification of existing schemes. Several numerical tests are performed to verify the ability of the proposed scheme to accurately capture small perturbations of steady states.

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.

Kinetic Flux Vector Splitting Method for Numerical Study of Two-dimensional Ripa Model

Abstract The kinetic flux vector splitting method has been introduced for two-dimensional system of shallow water equations with horizontal temperature gradients. The scheme preserves positivity conditions and resolves different regions of shock waves, rarefaction waves and contact discontinuity with negligible oscillations. The scheme is based on splitting of flux functions of the Ripa model. Moreover Runge-Kutta time stepping technique with MUSCL-type initial reconstruction is used to guarantee higher order accurate solution. The numerical example is taken from already published article. The obtained results reveal the accuracy and robustness of the proposed method.

An Entropy-Stable Residual Distribution Scheme for the System of Two-Dimensional Inviscid Shallow Water Equations

An entropy-stable residual distribution (RD) method is developed for the systems of two-dimensional shallow water equations (SWE). The construction of entropy stability for the residual distribution method is derived from finite volume method principles, albeit using a multidimensional approach. The paper delves into in-depth discussions on how finite volume methods achieve entropy stability in a “one-dimensional” sense for two-dimensional systems of SWE unlike the residual distribution methods. Results herein demonstrate the superiority of the entropy-stable RD methods relative to their finite volume counterparts, especially on highly irregular triangular grids. The comparative results with other established RD methods are also included, depicting similar performances with the Lax–Wendroff method for unsteady smooth flows but more accurate than the multidimensional upwind approaches (N, LDA) on both smooth and discontinuous test cases.