适用于间断Galerkin方法的限制器研究

开发了一种适用于高精度间断Galerkin方法的斜率（多项式系数）限制器。与现有的斜率限制器不同,该限制器实施过程不考虑网格单元类型（三角形或四边形）,通过全微分方法构造新的多项式系数,因此,该限制器能够适用于各种类型网格——结构化网格、具有单一单元的非结构化网格和具有混合单元的非结构化网格。由于该限制器能够方便地应用于具有混合单元的非结构化网格,因此,本文使用的程序能够方便地求解具有复杂几何结构的流动问题。本文利用一些典型算例对其性能进行了验证,表明该限制器适用于不同类型的网格单元,能够在光滑解区保证高的精度,并能够在阊断区抑帛3非物理振荡。     

Lattice Boltzmann modeling of shallow water flows over discontinuous beds

Numerical modeling of shallow water flows over discontinuous beds is presented. The flows are described with the shallow water equations and the equations are solved using the lattice Boltzmann method (LBM) with single relaxation time (Bhatnagar–Gross–Krook‐LBM (BGK‐LBM)) and the multiple relaxation time (MRT‐LBM). The weighted centered scheme for force term together with the bed height for a bed slope is described to improve simulation of flows over discontinuous bed. Furthermore, the resistance stress is added to include the local head loss caused by flow over a step. Four test cases, one‐dimensional tidal over regular bed and steps, dam‐break flows, and two‐dimensional shallow water flow over a square block, are considered to verify the present method. Agreements between predictions and analytical solutions are satisfactory. Furthermore, the performance and CPU cost time of BGK‐LBM and MRT‐LBM are compared and studied. The results have shown that the lattice Boltzmann method is simple and accurate for simulating shallow water flows over discontinuous beds. This demonstrates the capability and applicability of the lattice Boltzmann method in modeling shallow water flows on bed topography with a discontinuity in practical hydraulic engineering. Copyright © 2014 John Wiley & Sons, Ltd.     

Godunov-type solution of the shallow water equations on adaptive unstructured triangular grids

A Godunov-type upwind finite volume solver of the non-linear shallow water equations is described. The shallow water equations are expressed in a hyperbolic conservation law formulation for application to cases where the bed topography is spatially variable. Inviscid fluxes at cell interfaces are computed using Roe's approximate Riemann solver. Second-order accurate spatial calculations of the fluxes are achieved by enhancing the polynomial approximation of the gradients of conserved variables within each cell. Numerical oscillations are curbed by means of a non-linear slope limiter. Time integration is second-order accurate and implicit. The numerical model is based on dynamically adaptive unstructured triangular grids. Test cases include an oblique hydraulic jump, jet-forced flow in a flat-bottomed circular reservoir, wind-induced circulation in a circular basin of non-uniform bed topography and the collapse of a circular dam. The model is found to give accurate results in comparison with published analytical and alternative numerical solutions. Dynamic grid adaptation and the use of a second-order implicit time integration scheme are found to enhance the computational efficiency of the model.     

Entropy dissipation scheme and minimums‐increase‐and‐maximums‐decrease slope limiter

In this paper, we continue to study the entropy dissipation scheme developed in former. We start with a numerical study of the scheme without the entropy dissipation term on the linear advection equation, which shows that the scheme is stable and numerical dissipation and numerical dispersion free for smooth solutions. However, the numerical results for discontinuous solutions show nonlinear instabilities near jump discontinuities. This is because the scheme enforces two related conservation properties in the computation. With this study, we design a so‐called ‘minimums‐increase‐and‐maximums‐decrease’ slope limiter in the reconstruction step of the scheme and delete the entropy dissipation in the linear fields and reduce the entropy dissipation terms in the nonlinear fields. Numerical experiments show improvements of the designed scheme compared with the results presented in former. However, the minimums‐increase‐and‐maximums‐decrease limiter is still not perfect yet, and better slope limiters are still sought. Copyright © 2011 John Wiley & Sons, Ltd.     

An adaptive discretization of shallow‐water equations based on discontinuous Galerkin methods

In this paper, we present a discontinuous Galerkin formulation of the shallow‐water equations. An orthogonal basis is used for the spatial discretization and an explicit Runge–Kutta scheme is used for time discretization. Some results of second‐order anisotropic adaptive calculations are presented for dam breaking problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities like hydraulic jumps. Copyright © 2006 John Wiley & Sons, Ltd.     

A discontinuous Galerkin scheme for the numerical solution of flow problems with discontinuities

In this paper, a new discontinuous Galerkin finite element method for the numerical solution of flow problems with discontinuities is presented. The method is based on the limitation in every cell of the difference between the extrema values and the mean value of the numerical solution. The algorithm and technical details for the implementation of the method are presented in one‐and two‐dimensional problems. Numerical experiments for classical test problems are solved on unstructured triangulations to demonstrate the performance of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

Two‐dimensional dam break flow simulation

Numerical modelling of shallow water flow in two dimensions is presented in this work with the results obtained in dam break tests. Free surface flow in channels can be described mathematically by the shallow‐water system of equations. These equations have been discretized using an approach based on unstructured Delaunay triangles and applied to the simulation of two‐dimensional dam break flows. A cell centred finite volume method based on Roe's approximate Riemann solver across the edges of the cells is presented and the results are compared for first‐ and second‐order accuracy. Special treatment of the friction term has been adopted and will be described. The scheme is capable of handling complex flow domains as shown in the simulation corresponding to the test cases proposed, i.e. that of a dam break wave propagating into a 45° bend channel (UCL) and in a channel with a constriction (LNEC‐IST). Comparisons of experimental and numerical results are shown. Copyright © 2000 John Wiley & Sons, Ltd.     

An unstructured finite‐volume method for coupled models of suspended sediment and bed load transport in shallow‐water flows

The aim of this work is to develop a well‐balanced finite‐volume method for the accurate numerical solution of the equations governing suspended sediment and bed load transport in two‐dimensional shallow‐water flows. The modelling system consists of three coupled model components: (i) the shallow‐water equations for the hydrodynamical model; (ii) a transport equation for the dispersion of suspended sediments; and (iii) an Exner equation for the morphodynamics. These coupled models form a hyperbolic system of conservation laws with source terms. The proposed finite‐volume method consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The gradient fluxes are discretized using a modified Roe's scheme using the sign of the Jacobian matrix in the coupled system. A well‐balanced discretization is used for the treatment of source terms. In this paper, we also employ an adaptive procedure in the finite‐volume method by monitoring the concentration of suspended sediments in the computational domain during its transport process. The method uses unstructured meshes and incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep sediment concentrations and bed load gradients that may form in the approximate solutions. Details are given on the implementation of the method, and numerical results are presented for two idealized test cases, which demonstrate the accuracy and robustness of the method and its applicability in predicting dam‐break flows over erodible sediment beds. The method is also applied to a sediment transport problem in the Nador lagoon.Copyright © 2013 John Wiley & Sons, Ltd.     

Weighted essential non‐oscillatory schemes for tidal bore on unstructured meshes

In this paper, the third‐order weighted essential non‐oscillatory (WENO) schemes are used to simulate the two‐dimensional shallow water equations with the source terms on unstructured meshes. The balance of the flux and the source terms makes the shallow water equations fit to non‐flat bottom questions. The simulation of a tidal bore on an estuary with trumpet shape and Qiantang river is performed; the results show that the schemes can be used to simulate the current flow accurately and catch the stronger discontinuous in water wave, such as dam break and tidal bore effectively. Copyright © 2008 John Wiley & Sons, Ltd.     

A well‐balanced explicit/semi‐implicit finite element scheme for shallow water equations in drying–wetting areas

Numerical solutions of the shallow water equations can be used to reproduce flow hydrodynamics occurring in a wide range of regions. In hydraulic engineering, the objectives include the prediction of dam break wave propagation, fluvial floods and other catastrophic flooding phenomena, the modeling of estuarine and coastal circulations, and the design and optimization of hydraulic structures. In this paper, a well‐balanced explicit and semi‐implicit finite element scheme for shallow water equations over complex domains involving wetting and drying is proposed. The governing equations are discretized by a fractional finite element method using a two‐step Taylor–Galerkin scheme. First, the intermediate increment of conserved variable is obtained explicitly neglecting the pressure gradient term. This is then corrected for the effects of pressure once the pressure increment has been obtained from the Poisson equation. In order to maintain the ‘well‐balanced’ property, the pressure gradient term and bed slope terms are incorporated into the Poisson equation. Moreover, a local bed slope modification technique is employed in drying–wetting interface treatments. The proposed model is well validated against several theoretical benchmark tests. Copyright © 2014 John Wiley & Sons, Ltd.     

An unstructured finite volume model for dam‐break floods with wet/dry fronts over complex topography

A robust, well‐balanced, unstructured, Godunov‐type finite volume model has been developed in order to simulate two‐dimensional dam‐break floods over complex topography with wetting and drying. The model is based on the nonlinear shallow water equations in hyperbolic conservation form. The inviscid fluxes are calculated using the HLLC approximate Riemann solver and a second‐order spatial accuracy is achieved by implementing the MUSCL reconstruction technique. To prevent numerical oscillations near shocks, slope‐limiting techniques are used for controlling the total variation of the reconstructed field. The model utilizes an explicit two‐stage Runge–Kutta method for time stepping, whereas implicit treatments for friction source terms. The novelties of the model include the flux correction terms and the water depth reconstruction method both for partially and fully submerged cells, and the wet/dry front treatments. The proposed flux correction terms combined with the water depth reconstruction method are necessary to balance the bed slope terms and flux gradient in the hydrostatical steady flow condition. Especially, this well‐balanced property is also preserved in partially submerged cells. It is found that the developed wet/dry front treatments and implicit scheme for friction source terms are stable. The model is tested against benchmark problems, laboratory experimental data, and realistic application related to dam‐break flood wave propagation over arbitrary topography. Numerical results show that the model performs satisfactorily with respect to its effectiveness and robustness and thus has bright application prospects. Copyright © 2010 John Wiley & Sons, Ltd.     

Adaptive Q‐tree Godunov‐type scheme for shallow water equations

This paper presents details of a second‐order accurate, Godunov‐type numerical model of the two‐dimensional shallow water equations (SWEs) written in matrix form and discretized using finite volumes. Roe's flux function is used for the convection terms and a non‐linear limiter is applied to prevent unwanted spurious oscillations. A new mathematical formulation is presented, which inherently balances flux gradient and source terms. It is, therefore, suitable for cases where the bathymetry is non‐uniform, unlike other formulations given in the literature based on Roe's approximate Riemann solver. The model is based on hierarchical quadtree (Q‐tree) grids, which adapt to inherent flow parameters, such as magnitude of the free surface gradient and depth‐averaged vorticity. Validation tests include wind‐induced circulation in a dish‐shaped basin, two‐dimensional frictionless rectangular and circular dam‐breaks, an oblique hydraulic jump, and jet‐forced flow in a circular reservoir. Copyright © 2001 John Wiley & Sons, Ltd.     

The composite finite volume method on unstructured meshes for the two‐dimensional shallow water equations

A composite finite volume method (FVM) is developed on unstructured triangular meshes and tested for the two‐dimensional free‐surface flow equations. The methodology is based on the theory of the remainder effect of finite difference schemes and the property that the numerical dissipation and dispersion of the schemes are compensated by each other in a composite scheme. The composite FVM is formed by global composition of several Lax–Wendroff‐type steps followed by a diffusive Lax–Friedrich‐type step, which filters out the oscillations around shocks typical for the Lax–Wendroff scheme. To test the efficiency and reliability of the present method, five typical problems of discontinuous solutions of two‐dimensional shallow water are solved. The numerical results show that the proposed method, which needs no use of a limiter function, is easy to implement, is accurate, robust and is highly stable. Copyright © 2001 John Wiley & Sons, Ltd.     

High-order discontinuous Galerkin solver on hybrid anisotropic meshes for laminar and turbulent simulations

Efficient and robust solution strategies are developed for discontinuous Galerkin （DG） discretization of the Navier-Stokes （NS） and Reynolds-averaged NS （RANS） equations on structured/unstructured hybrid meshes. A novel line-implicit scheme is devised and implemented to reduce the memory gain and improve the computational eificiency for highly anisotropic meshes. A simple and effective technique to use the mod- ified Baldwin-Lomax （BL） model on the unstructured meshes for the DC methods is proposed. The compact Hermite weighted essentially non-oscillatory （HWENO） limiters are also investigated for the hybrid meshes to treat solution discontinuities. A variety of compressible viscous flows are performed to examine the capability of the present high- order DG solver. Numerical results indicate that the designed line-implicit algorithms exhibit weak dependence on the cell aspect-ratio as well as the discretization order. The accuracy and robustness of the proposed approaches are demonstrated by capturing com- plex flow structures and giving reliable predictions of benchmark turbulent problems.     

Assessment of shock capturing schemes for discontinuous Galerkin method

This paper carries out systematical investigations on the performance of several typical shock-capturing schemes for the discontinuous Galerkin （DG） method, including the total variation bounded （TVB） limiter and three artificial diffusivity schemes （the basis function-based （BF） scheme, the face residual-based （FR） scheme, and the element residual-based （ER） scheme）. Shock-dominated flows （the Sod problem, the Shu- Osher problem, the double Mach reflection problem, and the transonic NACA0012 flow） are considered, addressing the issues of accuracy, non-oscillatory property, dependence on user-specified constants, resolution of discontinuities, and capability for steady solutions. Numerical results indicate that the TVB limiter is more efficient and robust, while the artificial diffusivity schemes are able to preserve small-scale flow structures better. In high order cases, the artificial diffusivity schemes have demonstrated superior performance over the TVB limiter.     

Anisotropic slope limiting for discontinuous Galerkin methods

In this paper, we present an anisotropic version of a vertex‐based slope limiter for discontinuous Galerkin methods. The limiting procedure is carried out locally on each mesh element utilizing the bounds defined at each vertex by the largest and smallest mean value from all elements containing the vertex. The application of this slope limiter guarantees the preservation of monotonicity. Unnecessary limiting of smooth directional derivatives is prevented by constraining the x and y components of the gradient separately. As an inexpensive alternative to optimization‐based methods based on solving small linear programming problems, we propose a simple operator splitting technique for calculating the correction factors for the x and y derivatives. We also provide the necessary generalizations for using the anisotropic limiting strategy in an arbitrary rotated frame of reference and in the vicinity of exterior boundaries with no Dirichlet information. The limiting procedure can be extended to elements of arbitrary polygonal shape and three dimensions in a straightforward fashion. The performance of the new anisotropic slope limiter is illustrated by two‐dimensional numerical examples that employ piecewise linear discontinuous Galerkin approximations. Copyright © 2017 John Wiley & Sons, Ltd.     

Weighted average flux method and flux limiters for the numerical simulation of shock waves in rigid porous media

The one‐dimensional flow field generated by the passage of a shock wave in a rigid, thermoelastic porous foam has been simulated using a two‐phase mathematical model. The work presented here makes use of the weighted average flux method to solve the system of six equations that govern the problem. Spurious oscillations are eliminated through the application of total variation diminishing limiting methods. Four different limiters were tested: van Leer, SuperA, MinA and van Albada. Numerical tests were carried out to verify the performance of each flux limiter in terms of accuracy. The results were compared to analytical and previously obtained data to assess the performance of the mathematical model. Excellent agreement was obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

HLLC scheme with novel wave‐speed estimators appropriate for two‐dimensional shallow‐water flow on erodible bed

This paper presents a first‐order HLLC (Harten‐Lax‐Van Leer with contact discontinuities) scheme to solve the Saint‐Venant shallow‐water equations, including morphological evolution of the bed by erosion and deposition of sediments. The Exner equation is used to model the morphological evolution of the bed, while a closure equation is needed to evaluate the rate of sediment transport. The system of Saint‐Venant–Exner equations is solved in a fully coupled way using a finite‐volume technique and a HLLC solver for the fluxes, with a novel wave‐speed estimator adapted to the Exner equation. Wave speeds are usually derived by computing the eigenvalues of the full system, which is highly time‐consuming when no analytical expression is available. In this paper, an eigenvalue analysis of the full system is conducted, leading to simple but still accurate wave‐speed estimators. The new numerical scheme is then tested in three different situations: (1) a circular dam‐break flow over movable bed, (2) an one‐dimensional bed aggradation problem simulated on a 2D unstructured mesh and (3) the case of a dam‐break flow in an erodible channel with a sudden enlargement, for which experimental measurements are available. Copyright © 2010 John Wiley & Sons, Ltd.     

A stabilization for three‐dimensional discontinuous Galerkin discretizations applied to nonhydrostatic atmospheric simulations

A discontinuous Galerkin nonhydrostatic atmospheric model is used for two‐dimensional and three‐dimensional simulations. There is a wide range of timescales to be dealt with. To do so, two different implicit/explicit time discretizations are implemented. A stabilization, based upon a reduced‐order discretization of the gravity term, is introduced to ensure the balance between pressure and gravity effects. While not affecting significantly the convergence properties of the scheme, this approach allows the simulation of anisotropic flows without generating spurious oscillations, as it happens for a classical discontinuous Galerkin discretization. This approach is shown to be less diffusive than usual spatial filters. A stability analysis demonstrates that the use of this modified scheme discards the instability associated with the usual discretization. Validation against analytical solutions is performed, confirming the good convergence and stability properties of the scheme. Numerical results demonstrate the attractivity of the discontinuous Galerkin method with implicit/explicit time integration for large‐scale atmospheric flows. Copyright © 2015 John Wiley & Sons, Ltd.     

A γ‐DGBGK scheme for compressible multi‐fluids

The paper presents a Discontinuous Galerkin γ‐BGK (γ‐DGBGK) method for compressible multicomponent flow simulations by coupling the discontinuous Galerkin method with a γ‐BGK scheme based on WENO limiters. In this γ‐DGBGK method, the construction of the flux in the DG method is based on the kinetic scheme which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous terms in the flux formulation at cell interfaces. WENO limiters are used to obtain uniform high‐order accuracy and sharp non‐oscillatory shock transition, and time accuracy obtained by integration for the flux function at the cell interface. Numerical examples in one and two space dimensions are presented to illustrate the robust and accuracy of the present scheme. Copyright © 2010 John Wiley & Sons, Ltd.