Scheme For Shallow Water Equations Research Articles

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

Об одной безусловно устойчивой схеме класса КАБАРЕ для системы уравнений мелкой воды

В работе рассматривается новый подход к построению неявных безусловно устойчивых схем в рамках балансно-характеристической методики КАБАРЕ применительно к системе уравнений мелкой воды. Метод основан на идее инверсии координатных осей в явной схеме КАБАРЕ для преодоления ограничения на шаг по времени. Система уравнений является нелинейной, так как в уравнения включены минмаксные операции лимитирования на основе принципа максимума для локальных инвариантов Римана. Такое лимитирование существенно улучшает дисперсионные свойства схемы. Нелинейная система уравнений решается методом бегущего счета. В работе представлен вывод схемы для числа Куранта–Фридрихса–Леви CFL ≤ 1 и CFL > 1. Приведены тестовые расчеты на примерах простейшего уравнения переноса и одномерных задачах мелкой воды для дозвукового случая. Сделаны выводы о влиянии нелинейной коррекции потоков на решение для данной схемы. The paper considers a new approach to the construction of implicit unconditionally stable schemes within the framework of the CABARET balance-characteristic technique in relation to the system of shallow water equations. The method is based on the idea of inverting coordinate axes in the CABARET scheme to overcome the time step limitation. The system of equations is nonlinear, since the equations include minmax limiting operations based on the maximum principle for local Riemann invariants. This limitation significantly improves the dispersion properties of the numerical scheme. The nonlinear system of equations is solved using the marching order method. The paper presents the derivation of the numerical scheme for the Courant–Friedrichs–Lewy number CFL ≤ 1 and CFL > 1. Test calculations are presented on the simplest transport equation and one-dimensional shallow water problems for the subsonic case. Conclusions are drawn about the influence of nonlinear flux correction on the solution.

An Implicit Quadrature-Free Modal Discontinuous Galerkin (DG) Scheme for Shallow Water Equations

An Implicit Quadrature-Free Modal Discontinuous Galerkin (DG) Scheme for Shallow Water Equations

A well-balanced positivity-preserving multidimensional central scheme for shallow water equations

Recently, an efficient and multidimensional central scheme has been proposed by applying the limiting process diagonally (Yan et al. (2023) [25]). The aforementioned approach is applied to solve the two-dimensional shallow water equations with non-flat bottom topography in this study. The nonnegativity of the updated water depth is preserved under a reasonable CFL condition. The well-balancedness is achieved by simply applying central difference discretization to the source term using the surface gradient method. The conservation of water mass is obtained by changing the scheme into conservation form. All the key features of the present scheme have been rigorously proven. Several numerical experiments are provided to illustrate the robustness of the proposed scheme, including simulations of shallow water equations with Manning's bottom friction and Coriolis forces.

Hybrid fifth-order unequal-sized weighted essentially non-oscillatory scheme for shallow water equations

Hybrid fifth-order unequal-sized weighted essentially non-oscillatory scheme for shallow water equations

A New Well-Balanced Finite Volume CWENO Scheme for Shallow Water Equations over Bottom Topography

A New Well-Balanced Finite Volume CWENO Scheme for Shallow Water Equations over Bottom Topography

A mass conservative, well balanced and positivity-preserving central scheme for shallow water equations

Based on the invariant-region-preserving (IRP) reconstruction method introduced in [Yan, Tong and Chen, Appl. Math. Comput., 436 (2023) 127500], a second order unstaggered central scheme is proposed to solve the shallow water equations with bottom topography in the framework that the bottom is discretized by a continuous, piecewise linear approximation. The reconstruction applies a modification locally on a preliminary reconstructed surface gradient in every cell to yield a convexity property of the sampled point value in the forward and backward projections. The water mass conservation is proved by rewriting the scheme in a conservation form. The modification does not change the preliminary reconstructed slope of water surface for the lake-at-rest steady state and then keeps the well-balancing property of the surface gradient method. The convexity property ensures the nonnegativity of the updated water depth under a large CFL number which yields a considerable speed-up. The numerical experiments are shown to demonstrate the robustness of the scheme.

Well balanced finite volume schemes for shallow water equations on manifolds

In this paper we propose a novel second-order accurate well balanced scheme for shallow water equations in general covariant coordinates over manifolds. In our approach, once the gravitational field is defined for the specific case, one equipotential surface is detected and parametrized by a frame of general covariant coordinates. This surface is the manifold whose covariant parametrization induces a metric tensor. The model is then re-written in a hyperbolic form with a tuple of conserved variables composed both of the evolving physical quantities and the metric coefficients. This formulation allows the numerical scheme to automatically compute the curvature of the manifold as long as the physical variables are evolved. In a classical well balanced formulation, the knowledge of a given equilibrium is exploited for evolving the specific state seen as the sum of the equilibrium and the fluctuations of the state around the equilibrium itself. On the contrary, the numerical approach proposed here is automatically well balanced for the water at rest solution (defined by zero velocity and constant free surface) for general manifoldswithout having to exploit the exact equilibrium profile during the computation. In particular, this numerical strategy allows to preserve the accuracy of the water at rest equilibrium at machine precision and on large timescales, even for non-smooth bottom topographies. As a matter of fact, the proposed local polynomial reconstruction, needed at each time step by the scheme for evolving the state, is built in order to automatically cancel any numerical error committed for describing jumps in the bathymetry. As a further effect, also out of the equilibrium, the typical spurious non-physical oscillations of the recovered numerical solution in a neighborhood of the discontinuities are healed. Thus, once the information on both the bathymetry and the metric of the manifold are properly collected in the flux terms and in the nonconservative products, the resulting numerical strategy turns out to be more accurate in finding also non-equilibrium solutions. Numerical results close the work. After having proved that the scheme is second-order accurate, we test the recovery of the water at rest equilibrium at machine precision for different bottom topographies (eventually discontinuous) with various metric on very long simulation times. In particular, after having considered smooth bathymetries with compact support, we relax the assumption on the continuity of water at rest equilibrium with discontinuous bathymetries disturbed by a white noise.

Adaptive physical-constraints-preserving unstaggered central schemes for shallow water equations on quadrilateral meshes

A well-balanced and positivity-preserving adaptive unstaggered central scheme for two-dimensional shallow water equations with nonflat bottom topography on irregular quadrangles is presented. The irregular quadrilateral mesh adds to the difficulty of designing unstaggered central schemes. In particular, the integral of the source term needs to subtly be dealt with. A new method of discretizing the source term for the well-balanced property is proposed, which is one of the main contributions of this work. The spacial second-order accuracy is obtained by constructing piecewise bilinear functions. Another novelty is that we introduce a strong-stability-preserving Unstaggered-Runge–Kutta method to improve the accuracy in time integration. Adaptive moving mesh strategies are introduced to couple with the current unstaggered central scheme. The well-balanced property is still valid. The positivity-preserving property can be proved when the cells shrink. We prove that the current adaptive unstaggered central scheme can obtain the stationary solution (“lake at rest” steady solutions) and guarantee the water depth to be nonnegative. Several classical problems of shallow water equations are shown to demonstrate the properties of the current numerical scheme.

Surface reconstruction schemes for shallow water equations with a nonconservative product source term

We aim to introduce surface reconstruction (SR) schemes for shallow water equations with a nonconservative product source term. The SR scheme is used to define the intermediate water depth and the bottom topography on the cell boundaries. The key ingredient of the SR scheme is to smooth the water surface level or the bottom topography while maintaining their the monotone property. The discretization of the integral of the nonconservative product term is established with the aid of a family of pathes chosen in the phase space. The discretized source term is closer to the exact one. For obtaining a second-order accuracy, we introduce a non-oscillatory monotone-preserving reconstruction method. We establish the conditions of the SR scheme for obtaining the semi-discrete and fully-discrete entropy inequalities and prove that the introduced SR scheme can maintain the stationary solutions and guarantee the water depth to be nonnegative. Several numerical results of the one- and two-dimensional shallow water equations confirm that the SR scheme can preserve the stationary state, the positivity of the water depth, and is efficient when computing the shallow water flows over a step.

Positivity-Preserving and Well-Balanced Adaptive Surface Reconstruction Schemes for Shallow Water Equations with Wet-Dry Fronts

Positivity-Preserving and Well-Balanced Adaptive Surface Reconstruction Schemes for Shallow Water Equations with Wet-Dry Fronts

A well-balanced weighted compact nonlinear scheme for shallow water equations on curvilinear grids

Shallow water equations have important applications in civil engineering. For these balance models, a numerical scheme with a well-balanced property is useful for reducing numerical errors and hence for resolving small perturbations of steady state solutions. In the past two decades, some high-order well-balanced finite difference schemes have been developed for the shallow water equations on Cartesian grids. However, it is not clear whether the well-balanced property can be maintained on curvilinear grids, which are often used in practice for physical domains with curved boundaries. In this work, we consider a two-dimensional case to show that the weighted compact nonlinear scheme is well-balanced for the shallow water equations in a pre-balanced form provided that geometric conservation laws are satisfied. Theoretical analysis and several representative numerical tests are conducted to validate the proposed fifth-order scheme.

A fully well-balanced scheme for shallow water equations with Coriolis force

The present work is devoted to the derivation of a fully well-balanced and positivepreserving numerical scheme for the shallow water equations with Coriolis force. The first main issue consists in preserving all the steady states, including the geostrophic equilibrium. Our strategy relies on a Godunov-type scheme with suitable source term and steady state discretisations. The second challenge lies in improving the order of the scheme while preserving the fully well-balanced property. A modification of the classical methods is required since no conservative reconstruction can preserve all the steady states in the case of rotating shallow water equations. A steady state detector is used to overcome this matter. Some numerical experiments are presented to show the relevance and the accuracy of both first-order and second-order schemes.

Godunov-type large time step scheme for shallow water equations with bed-slope source term

This paper presents a Godunov-type large time step (LTS) solver of the non-homogeneous shallow water equations (SWEs). Source terms are decomposed into simple characteristic waves in approximate Riemann solvers (ARS) and exact Riemann solvers (ERS), and information is transferred over multiple cells per time step using the LTS method. Benchmark simulations are presented using different solution algorithms (ARS and ERS with and without entropy fixes) for two rarefactions driven by divergent flow, a pair of bores driven by opposing flows, and a dam break over a shelf-like step. In these cases, spurious flow discontinuities and oscillations can occur for Courant–Friedrichs–Lewy number (CFL) > 1 in the absence of an entropy fix. Implementation of a weak-solution LTS entropy fix improves the results, but shock shifting nevertheless occurs in certain cases. The paper also considers steady, frictionless, transcritical flow over a bed hump. In this final case, the model is run for integer CFL ranging from 1 to 10. For CFL ≤ 3, satisfactory results are obtained (without divergence and oscillation) using ARS without an entropy fix. For larger CFL, the results either diverge or exhibit convergent oscillations downstream of the hydraulic jump. Use of an entropy fix designed for implementation in an LTS scheme improves the results for CFL ≤ 5.

A well-balanced high-order scheme on van Leer-type for the shallow water equations with temperature gradient and variable bottom topography

A well-balanced high-order scheme for shallow water equations with variable topography and temperature gradient is constructed. This scheme is of van Leer-type and is based on exact Riemann solvers. The scheme is shown to be able to capture almost exactly the stationary smooth solutions as well as stationary elementary discontinuities. Numerical tests show that the scheme gives a much better accuracy than the Godunov-type scheme and can work well even in the resonant regime. Wave interaction problems are also tested where the scheme possesses a good accuracy. It turns out that the superbee limiter can provide us with more accurate approximations than van Leer’s limiter.

Large Time Step TVD High Order Scheme for Shallow Water Equations

Large Time Step TVD High Order Scheme for Shallow Water Equations

High order well-balanced finite difference WENO interpolation-based schemes for shallow water equations

A numerical framework of the generalized form of high order well-balanced finite difference weighted essentially non-oscillatory (WENO) interpolation-based schemes is proposed for the shallow water equations. It demonstrates more flexible construction process than the classical WENO reconstruction-based schemes. The weighted compact nonlinear schemes and finite difference alternative WENO schemes are two specific cases. To maintain the exact C-property, the splitting technique for the source term in the finite difference scheme [Xing and Shu, J. Comput. Phys. 208 (2005)] and the reconstruction technique in the finite volume WENO scheme [Xing and Shu, J. Comput. Phys. 214 (2006)] are adopted. The proposed scheme can be proved mathematically to maintain the exact C-property and demonstrates numerically that it is well-balanced by construction for the stationary water surface. Moreover, the local characteristic projections are employed to further mitigate the Gibbs oscillations. The proposed generic high order WENO schemes not only achieve high order accuracy but also capture the high gradients/shock waves essentially non-oscillatory. Meanwhile, the small perturbation problems can be resolved well on a coarse grid.

Operator splitting based central-upwind schemes for shallow water equations with moving bottom topography

Operator splitting based central-upwind schemes for shallow water equations with moving bottom topography

Energy-stable staggered schemes for the Shallow Water equations

We focus on the development and analysis of staggered schemes for the two-dimensional non-linear Shallow Water equations with varying bathymetry. Semi-implicit and fully explicit time-discretizations are proposed. Particular attention is paid on non-linear stability results, principally considered here through discrete energy dissipation arguments. To address such an issue, specific convective fluxes are employed, implying diffusive terms relying on the pressure gradient. In addition of providing an explicit control of the discrete energy budget, the method is shown to preserve motionless steady states as well as the positivity of the water height. These properties are still satisfied in a fully explicit context, provided an appropriate discretization of the pressure gradient. These stability results make the approach particularly robust and efficient, for both coastal flows and low-Froude number regimes. As a result, in addition of a great ease of implementation, the presented schemes meet the operational requirements attached to the simulation of large and small scale oceanic flows.

Staggered Scheme for Shallow Water Equations with Quadtree-subgrid

To get accurate results, high-resolution numerical simulations are often required. However, computation using high resolution grid often require larger computer memory and longer computational time. One way to overcome this problem is by adopting a scheme that can handle a non-homogeneous grid; i.e. a computational domain with two or more resolutions. In this work, a two-level quadtree-subgrid method was applied. It uses a combination of structured grid with hierarchical quadtree ordering. This method is applied on a staggered grid scheme for the two-dimensional shallow water equations. A benchmark simulation of wave run-up on a conical island was conducted for validation. Comparison between computing time of simulations using the fine grid and the quadtree-subgrid shows the effectiveness of this quadtree-subgrid scheme.