Hydrostatic Reconstruction Research Articles

Semi-discrete entropy-preserving surface reconstruction schemes for the shallow water equations: Analysis of physical structures

We aim to introduce a new variant of hydrostatic reconstruction scheme, which is physical-structure-preserving and semi-discrete entropy-preserving for shallow water equations with a discontinuous bottom topography, based on novel surface reconstructions (NSR). The NSR is used to define approximate Riemann states with respect to the water depth and the velocity to compute consistent numerical fluxes across the cell interface and the upwind interface source term. This work is motivated by path-conservative surface reconstructions (Dong (2023) [19]), in which the conditions to satisfy the entropy inequality are complicated. We propose the NSR scheme to satisfy the entropy inequality without the corresponding complicated conditions. A key advantage of the NSR scheme is to preserve the physical structure of the water surface in the sense that when the water surface is continuous, then the defined approximate Riemann state should inherit this property. The physical-structure-preserving property is indispensable for obtaining a more accurate discretized source term and avoiding artificial “waterfall” effects. We prove that the NSR scheme satisfies a semi-discrete entropy condition and can guarantee the water depth to be nonnegative, and maintain the stationary state. We show several computed results to compare the NSR scheme with the hydrostatic reconstruction scheme based on subcell reconstructions and the path-conservative surface reconstruction scheme. We also extend the NSR scheme for pollutant transport in shallow water based on adaptive moving triangular meshes. Finally, several numerical results obtained by the adaptive NSR scheme for pollutant transport in shallow water are shown.

A fully well-balanced hydrodynamic reconstruction

Abstract The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. These moving steady states are solution to a nonlinear equation. We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme.

A novel efficient and robust treatment of the friction source term in 2D shallow water inundation models

The technological advancements of the last few decades have fostered the use of two-dimensional (2D) Shallow water Equations (SWE) flood simulations not only in academic research but also in practical real-world applications and territorial planning. The evaluation of flow resistance due to friction is crucial as it plays a relevant role in a variety of flow conditions. However, problems arise in dam-break and overland flow applications involving very small water depth because the time scale connected to the friction term may be much smaller than the hydrodynamic time scale. To cope with this issue, known as source term stiffness, an implicit treatment of the friction term is frequently adopted in Finite Volume (FV) numerical schemes, but there are cases (null roughness coefficient or null flow velocity) where a division by zero may occur during calculations, leading to a crash of the algorithm. Herein, we propose a reformulation of the implicit friction term that is general, more stable and computationally efficient (with a speed-up of approximately 3 times for the routine running the friction computation) than currently available approaches. The novel implicit method allows to manage special conditions with null roughness or null discharge, without resorting to ad-hoc thresholds, and can be easily implemented in existing schemes with pointwise friction treatment. The novel friction approach is implemented in a purposely simple FV numerical scheme (hydrostatic reconstruction to account for the bed slope terms, first-order accuracy in time and space) - to better focus on the friction term treatment - and it is validated against analytical, synthetic, laboratory and real-world case studies, showing promising capabilities.

Improved well-balanced AWENO schemes with hydrostatic reconstruction for the Euler equations under gravitational fields

The Euler equations under gravitational fields allow the hydrostatic equilibrium states, which requires that the numerical scheme of the system should also have this characteristic. In our previous work, a well-balanced finite difference conservative AWENO scheme has been constructed to preserve the isothermal equilibrium state accurately (Fu et al., Appl. Numer. Math., 180:1-15, 2022). However, the scheme fails to maintain more complicated hydrostatic equilibrium states, such as the isentropic equilibrium state. In this study, the improved well-balanced AWENO schemes with hydrostatic reconstruction are designed to maintain the general equilibrium states numerically. The key technologies to achieve the well-balanced property are rewriting the source term into an appropriate form based on the equilibrium equation, acting on the conserved variables with an affine-invariant WENO or linearized WENO interpolation operator, employing hydrostatic reconstruction methods to create auxiliary conserved variables, and modifying the numerical flux with auxiliary conserved variables and higher-order conserved terms based on pressure. The proposed well-balanced schemes maintain the equilibrium solution near the machine precision and capture small perturbations of the equilibrium state without spurious oscillations even at the coarse mesh. Furthermore, they can be applicable to the general hydrostatic equilibrium more than the isothermal equilibrium state. Several numerical experiments are tested to show that the developed well-balanced schemes perform well in maintaining equilibrium states and capturing small perturbations.

Computational study of numerical flux schemes for mesoscale atmospheric flows in a Finite Volume framework

Abstract We develop, and implement in a Finite Volume environment, a density-based approach for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations are used to describe non-hydrostatic atmospheric flow. The well-balancing of the approach is ensured by a local hydrostatic reconstruction updated in runtime during the simulation to keep the numerical error under control. To approximate the solution of the Riemann problem, we consider four methods: Roe-Pike, HLLC, AUSM+-up and HLLC-AUSM. We assess our density-based approach and compare the accuracy of these four approximated Riemann solvers using two classical benchmarks, namely the smooth rising thermal bubble and the density current.

An efficient interface hydrostatic reconstruction for the two-layer shallow flows with arbitrary wet-dry fronts

An efficient interface hydrostatic reconstruction for the two-layer shallow flows with arbitrary wet-dry fronts

Fifth-order well-balanced positivity-preserving finite difference AWENO scheme with hydrostatic reconstruction for hyperbolic chemotaxis models

We propose a fifth-order well-balanced positivity-preserving finite difference alternative WENO (AWENO) scheme with the affine-invariant WENO interpolation based on the Z-type nonlinear weights for the hyperbolic chemotaxis models. By using the techniques of source term reformulation, hydrostatic reconstruction of the interpolated conservative variables and modification of the numerical fluxes, the finite difference discretization is fifth-order and well-balanced. Moreover, the first-order interpolation with the Lax-Friedrichs (LF) flux and a reduced time step for the proposed discretized scheme has been shown to satisfy the density's positivity-preserving (PP) property. Thus, a simple positivity-preserving (PP) limiter conjugating the fifth-order hydrostatic reconstructed flux with the first-order positivity-preserving LF flux is introduced for extreme problems. Meanwhile, this improved approach strictly guarantees well-balanced property at the discrete level. Finally, one-, two-, and three-dimensional numerical examples are given to demonstrate the performance of the proposed AWENO scheme for this class of chemotaxis problems.

High-order fully well-balanced numerical methods for one-dimensional blood flow with discontinuous properties

In this paper, we are interested in the numerical study of the one-dimensional blood flow model with discontinuous mechanical and geometrical properties. We present the mathematical model together with its nondimensional form. We do an exhaustive investigation of all its stationary solutions and we propose high-order fully well-balanced numerical methods that are able to preserve all of them. They are based on the combination of the Generalized Hydrostatic Reconstruction and well-balanced reconstruction operators. These methods are able to deal with more than one discontinuous parameter. Several numerical tests are shown to prove its well-balanced and high-order properties, and its convergence to the exact solutions.

A unified surface-gradient and hydrostatic reconstruction scheme for the shallow water equations

We propose a new second-order accurate hydrostatic reconstruction scheme for the Saint-Venant system. Such a scheme needs to overcome several difficulties: besides the well-known issues of positivity and well-balancing there is also the difficulty of unphysical reflections from non-monotone bottom reconstructions. We address all of these problems at once by changing the logic of the reconstruction of the bottom, the water depth and the water surface level. The scheme gives excellent results in one and two space dimensions. To highlight the novel reconstruction of bottom and water surface, we call the scheme bottom-surface-gradient method (BSGM).

High-order well-balanced and positivity-preserving finite-difference AWENO scheme with hydrostatic reconstruction for shallow water equations

The shallow water equations (SWEs) admit still water steady-state solutions in which the flux gradients are exactly balanced by the source term. Furthermore, the no-water dry areas make the numerical simulation of the SWEs more difficult since the water height often loses positivity near the dry areas. In this study, we design a high-order well-balanced and positivity-preserving finite-difference AWENO scheme for solving the SWEs with or without dry areas. In this numerical framework, the hydrostatic reconstruction (HR) method is used with two main advantages: 1) The well-balanced property is achieved using the HR method with arbitrary monotone fluxes and the reformulated source terms. 2) The first-order scheme with the Lax-Friedrichs (LF) flux and the HR method maintains the positivity of the water height with an appropriate time step. Therefore, a simple positivity-preserving limiter conjugating the high-order reconstructed flux with the first-order positivity-preserving LF flux is introduced. One- and two-dimensional classical examples with or without dry areas demonstrate that the proposed scheme is high-order, well-balanced, and positivity-preserving.

A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of an update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.

High order well-balanced conservative finite difference AWENO scheme with hydrostatic reconstruction for the Euler equations under gravitational fields

The system of compressible Euler equations under gravitational fields drew much attention during the past few decades, due to its broad applications in astrophysics and atmospheric sciences. A critical feature of the system is that it admits a hydrostatic equilibrium state when the gradient of fluid flux is exactly balanced by the source term brought by gravitational fields, which is referred to as well-balancedness. In this paper, a well-balanced high order conservative finite difference alternative WENO (AWENO) scheme is proposed for the system. The finite difference AWENO scheme allows us to use an arbitrary consistent Riemann solver, such as LF, HLLC, etc. To preserve hydrostatic equilibrium, the hydrostatic reconstruction technique is adopted. We slightly modify the original hydrostatic reconstruction procedure to achieve the fifth order of accuracy in the framework of finite difference methods, without introducing spurious oscillations. Besides, the discretization is fully conservative without the source term, which is not the case for the methods with original hydrostatic reconstructions. Moreover, we will show that the well-balancedness is exactly guaranteed by our approach at the discrete level. Finally, numerical examples illustrate the effectiveness and robustness of the designed approach.

A robust hybrid unstaggered central and Godunov-type scheme for Saint-Venant–Exner equations with wet/dry fronts

We aim to propose a robust hybrid unstaggered central and Godunov-type scheme based on hydrostatic reconstruction (HR) for Saint-Venant–Exner equations with wetting and drying transitions. The discretization of the bed slope source term is based on the HR method to preserve the still water steady-state solution. The hydrodynamic model described by the Saint-Venant system is numerically solved using the well-balanced unstaggered central scheme proposed in Dong and Li (2020). The morphodynamic model described by the Exner equation is numerically solved by using a Lax–Friedrichs method. In solving the morphodynamic model, we proposed a novel “cut-off” function to guarantee the nonlinear stability and preserve the stationary solution. The present scheme can exactly obtain the stationary solution and is capable of guaranteeing the positivity of the water depth. The key contributions of this work are the well-balanced property at the wet-dry fronts and the stability of the current scheme in solving two physical models that have relatively either strong or weak interactions. Finally, we use several classical problems of the system to demonstrate these properties.

A unified asymptotic preserving and well-balanced scheme for the Euler system with multiscale relaxation

The design and analysis of a unified asymptotic preserving (AP) and well-balanced scheme for the Euler Equations with gravitational and frictional source terms is presented in this paper. The asymptotic behaviour of the Euler system in the limit of zero Mach and Froude numbers, and large friction is characterised by an additional scaling parameter. Depending on the values of this parameter, the Euler system relaxes towards a hyperbolic or a parabolic limit equation. Standard Implicit–Explicit Runge–Kutta schemes are incapable of switching between these asymptotic regimes. We propose a time semi-discretisation to obtain a unified scheme which is AP for the two different limits. A further reformulation of the semi-implicit scheme can be recast as a fully-explicit method in which the mass update contains both hyperbolic and parabolic fluxes. A space–time fully-discrete scheme is derived using a finite volume framework. A hydrostatic reconstruction strategy, an upwinding of the sources at the interfaces, and a careful choice of the central discretisation of the parabolic fluxes are used to achieve the well-balancing property for hydrostatic steady states. Results of several numerical case studies are presented to substantiate the theoretical claims and to verify the robustness of the scheme.

A Weakly Non-hydrostatic Shallow Model for Dry Granular Flows

A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the $\mu(I)$ rheology is considered. The model is obtained from an asymptotic analysis in a local reference system, where the non-hydrostatic contribution is supposed to be small compared to the hydrostatic one. The non-hydrostatic counterpart of the pressure may be written as the sum of two terms: one corresponding to the stress tensor and the other to the vertical acceleration. The model introduced here is weakly non-hydrostatic, in the sense that the non-hydrostatic contribution related to the stress tensor is not taken into account due to its complex implementation. A simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure, and it is based on a hydrostatic reconstruction. Two key points are: (i) the friction force has to be taken into account before solving the non-hydrostatic pressure. Otherwise, the incompressibility condition is not ensured; (ii) both the hydrostatic and the non-hydrostatic pressure are taken into account when dealing with the friction force. The model and numerical scheme are then validated based on several numerical tests, including laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models. An important result is that the simulated mass profiles up to the deposit and the front velocity are greatly improved. As expected, the influence of the non-hydrostatic pressure is shown to be larger for small values of the slope.

Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography

We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with Lipschitz continuous topography. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the space increment ∆x of the L 2 norm of the gradient of approximate solutions. By Diperna's method we conclude the strong convergence towards bounded weak entropy solutions.

A well-balanced ADER discontinuous Galerkin method based on differential transformation procedure for shallow water equations

This article develops a new discontinuous Galerkin (DG) method on structured meshes for solving shallow water equations. The method here applies the one-stage ADER (Arbitrary DERivatives in time and space) approach for the temporal discretization and employs the differential transformation procedure to express spatiotemporal expansion coefficients of the solution through low order spatial expansion coefficients recursively. Numerical fluxes using the hydrostatic reconstruction together with a simple source term discretization results in a well-balanced method accordingly. Compared with the Runge-Kutta DG (RKDG) methods, the proposed method needs less computer memory storage due to no intermediate stages. In comparison with traditional ADER schemes, this method avoids solving the generalized Riemann problems at inter-cells. The differential transformation procedure used here is cheaper than the Cauchy-Kowalewski procedure in traditional ADER schemes. In summary, this method is one-step, one-stage, fully-discrete, and easily achieves arbitrary high order accuracy in time and space. Theoretically as well as numerically, the proposed method is verified to be well-balanced. Numerical results illustrate the high order accuracy as well as good resolutions for discontinuous solutions. Moreover, this method is more efficient than the Runge-Kutta DG (RKDG) method.

A new second-order modified hydrostatic reconstruction for the shallow water flows with a discontinuous topography

We consider a new second-order hydrostatic reconstruction (HR) based on the bottom topography, the depth-averaged velocity, and the water surface level for the shallow water flows with discontinuous bottom topography. The Audusse's second-order hydrostatic reconstruction (HR) (Audusse et al. (2004) [1]) fails to correctly reflect the wave pattern for the large discontinuous bottom topography. We propose a new modified HR method to overcome this numerical difficulty. The well-balancing property is obtained by using the information of the neighboring cells which is used to modify the reconstruction at the wet-dry front. The positivity preserving property is ensured by a “draining” time-step technique. A number of classical problems are successfully solved by the new scheme. Especially, the new scheme yields superior results for the shallow downhill flow over steps.

Well‐balanced nonstaggered central schemes based on hydrostatic reconstruction for the shallow water equations with Coriolis forces and topography

The objective of this paper is to present a nonstaggered central scheme based on hydrostatic reconstruction (HR) for the shallow water equations with Coriolis forces and nonflat bottom topography. The discretization of the bed slope source term is based on the HR method to ensure the well‐balanced property for the stationary solution. We use the cell average of the discharge on the staggered cell to discretize the source term due to the effect of Coriolis forces. The positivity‐preserving property of the water depth is obtained by providing an appropriate CFL condition with the help of a “draining” time‐step technique. Finally, several two‐dimensional numerical results of classical problems of the system are presented to test these properties of the current scheme.

A 1D shallow-flow model for two-layer flows based on FORCE scheme with wet–dry treatment

Abstract The two-layer problem is defined as the coexistence of two immiscible fluids, separated by an interface surface. Under the shallow-flow hypothesis, 1D models are based on a four equations system accounting for the mass and momentum conservation in each fluid layer. Mathematically, the system of conservation laws modelling 1D two-layer flows has the important drawback of loss of hyperbolicity, causing that numerical schemes based on the eigenvalues of the Jacobian become unstable. In this work, well-balanced FORCE scheme is proposed for 1D two-layer shallow flows. The FORCE scheme combines the first-order Lax–Friedrichs flux and the second-order Lax–Wendroff flux. The scheme is supplemented with a hydrostatic reconstruction procedure in order to ensure the well-balanced behaviour of the model for steady flows even under wet–dry conditions. Additionally, a method to obtain high-accuracy numerical solutions for two-layer steady flows including friction dissipation is proposed to design reference benchmark tests for model validation. The enhanced FORCE scheme is faced to lake-at-rest benchmarking tests and steady flow cases including friction, demonstrating its well-balanced character. Furthermore, the numerical results obtained for highly unsteady two-layer dambreaks are used to analyse the robustness and accuracy of the model under a wide range of flow conditions.