Least-squares finite-element method for shallow-water equations with source terms

Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of source terms in momentum equations, such as the bottom slope and friction of bed, compounds the difficulties further. In this paper, a least-squares finite-element method for the space discretization and θ-method for the time integration is developed for the 2D non-conservative SWE including the source terms. Advantages of the method include： the source terms can be approximated easily with interpolation functions, no upwind scheme is needed, as well as the resulting system equations is symmetric and positive-definite, therefore, can be solved efficiently with the conjugate gradient method. The method is applied to steady and unsteady flows, subcritical and transcritical flow over a bump, 1D and 2D circular dam-break, wave past a circular cylinder, as well as wave past a hump. Computed results show good C-property, conservation property and compare well with exact solutions and other numerical results for flows with weak and mild gradient changes, but lead to inaccurate predictions for flows with strong gradient changes and discontinuities.     

Solution of the 2‐D shallow water equations with source terms in surface elevation splitting form

A vertex‐centred finite‐volume/finite‐element method (FV/FEM) is developed for solving 2‐D shallow water equations (SWEs) with source terms written in a surface elevation splitting form, which balances the flux gradients and source terms. The method is implemented on unstructured grids and the numerical scheme is based on a second‐order MUSCL‐like upwind Godunov FV discretization for inviscid fluxes and a classical Galerkin FE discretization for the viscous gradients and source terms. The main advantages are: (1) the discretization of SWE written in surface elevation splitting form satisfies the exact conservation property (??‐Property) naturally; (2) the simple centred‐type discretization can be used for the source terms; (3) the method is suitable for both steady and unsteady shallow water problems; and (4) complex topography can be handled based on unstructured grids. The accuracy of the method was verified for both steady and unsteady problems, including discontinuous cases. The results indicate that the new method is accurate, simple, and robust. Copyright © 2007 John Wiley & Sons, Ltd.     

An upwind compact mpact approach with group velocity control for compressible flow fields

In this paper we construct an upwind compact finite difference scheme with group velocity control for better simulation of compressible flow fields. Compared with traditional difference schemes, compact schemes have higher accuracy for the same stencil width. By means of the characteristic analysis of the operators, the group velocity of wave packets will be controlled to suppress the non‐physical oscillations in numerical solutions. In numerical simulation of the 3D compressible flow fields the third‐order accurate upwind compact operator is used to approximate the derivatives in the convection terms of the compressible N–S equations, the traditional finite difference scheme is used to approximate the viscous terms. Numerical solutions indicate that the method is satisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

A numerical method for 3D barotropic flows in turbomachinery

A numerical method for the simulation of 3D inviscid barotropic flows in rotating frames is presented. A barotropic state law incorporating a homogeneous-flow cavitation model is considered. The discretisation is based on a finite-volume formulation applicable to unstructured grids. A shock-capturing Roe-type upwind scheme is proposed for barotropic flows. The accuracy of the proposed method at low Mach numbers is ensured by ad-hoc preconditioning, preserving time consistency. An implicit time advancing only relying on the algebraic properties of the Roe flux function, and thus applicable to a variety of problems, is presented. The proposed numerical ingredients, already validated in a 1D context and applied to 3D non-rotating computations, are then applied to the 3D water flow around a typical turbopump inducer.     

Solution to Euler equations by high‐resolution upwind compact scheme based on flux splitting

A high‐resolution upwind compact method based on flux splitting is developed for solving the compressive Euler equations. The convective flux terms are discretized by using the modified advection upstream splitting method (AUSM). The developed scheme is used to compute the one‐dimensional Burgers equation and four different example problems of supersonic compressible flows, respectively. The results show that the high‐resolution upwind compact scheme based on modified AUSM+ flux splitting can capture shock wave and other discontinuities, obtain higher resolution and restrain numerical oscillation. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical modelling of shear-dependent mass transfer in large arteries

A numerical scheme for the simulation of blood flow and transport processes in large arteries is presented. Blood flow is described by the unsteady 3D incompressible Navier–Stokes equations for Newtonian fluids; solute transport is modelled by the advection–diffusion equation. The resistance of the arterial wall to transmural transport is described by a shear-dependent wall permeability model. The finite element formulation of the Navier–Stokes equations is based on an operator-splitting method and implicit time discretization. The streamline upwind/Petrov–Galerkin (SUPG) method is applied for stabilization of the advective terms in the transport equation and in the flow equations. A numerical simulation is carried out for pulsatile mass transport in a 3D arterial bend to demonstrate the influence of arterial flow patterns on wall permeability characteristics and transmural mass transfer. The main result is a substantial wall flux reduction at the inner side of the curved region. © 1997 John Wiley & Sons, Ltd.     

Time accurate local time stepping for the unsteady shallow water equations

Two time accurate local time stepping (LTS) strategies originally developed for the Euler equations are presented and applied to the unsteady shallow water equations of open channel flow. Using the techniques presented allows individual cells to be advanced to different points in time, in a time accurate fashion. The methods shown are incorporated into an explicit finite volume version of Roe's scheme which is implemented in conjunction with an upwind treatment for the source terms. A comparison is made between the results obtained using the conventional time stepping approach and the two LTS methods through a series of test cases. The results illustrate a number of benefits of using LTS which included reduced run times and improved solution accuracy. In addition it is shown how using an upwind source term treatment can be beneficial for flows dominated by the geometry. Copyright © 2005 John Wiley & Sons, Ltd.     

Concurrent use of finite element and finite volume methods for shallow water flows in locally 1‐D channel networks

A numerical model is developed for shallow water equation in locally 1‐D channel networks. The model concurrently uses the standard Galerkin finite element method for the continuity equation and the finite volume method with an upwind scheme for the momentum equation. The surface gradient method is consistently employed. A minimum treatment is given for channel junctions so that application to multiply connected channels do not require any special consideration The model is capable of computing different types of transcritical flows, wet and dry flows, and flows with complex source terms. Standardized test problems and laboratory experimental data are used for verifying the model. Applicability of the models is validated in a multiply connected channel network draining hydromorphic farmlands located in a West African savanna, and Manning's roughness coefficient is identified, so that the steady solution is consistent with field observations. Unsteady simulation demonstrates that the model is capable of stably reproducing shifts of hydraulic jumps in flows of sub‐millimeter water depths. Copyright © 2011 John Wiley & Sons, Ltd.     

极坐标系下Fourier-Legendre谱元方法与有限差分法数值扩散的比较

提出一种Fourier-Legendre谱元方法用于求解极坐标系下的Navier-Stokes方程,其中极点所在单元的径向采用Gauss-Radau积分点,避免了r=0处的1/r坐标奇异性。时间离散采用时间分裂法,引入数值同位素模型跟踪同位素的输运过程验证数值模拟的精度,分别利用谱元法和有限差分法的迎风差分格式求解匀速和加速坩埚旋转流动中的同位素方程。计算结果表明,有限差分法中的一阶迎风差分格式存在严重的数值假扩散,二阶迎风差分格式的数值结果较精确,增加节点可以有效地缓解数值扩散。然而,谱元法具有以较少节点得到高精度解的优势。     

A mixture differential quadrature method for solving two-dimensional incompressible Navier-Stokes equations

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A robust low‐dissipation AUSM‐family scheme for numerical shock stability on unstructured grids

The simple low‐dissipation advection upwind splitting method (SLAU) scheme is a parameter‐free, low‐dissipation upwind scheme that has been applied in a wide range of aerodynamic numerical simulations. In spite of its successful applications, the SLAU scheme could be showing shock instabilities on unstructured grids, as many other contact resolved upwind schemes. Therefore, a hybrid upwind flux scheme is devised for improving the shock stability of SLAU scheme, without compromising on accuracy and low Mach number performance. Numerical flux function of the hybrid scheme is written in a general form, in which only the scalar dissipation term is different from that of the SLAU scheme. The hybrid dissipation term is defined by using a differentiable multidimensional‐shock‐detection pressure weight function, and the dissipation term of SLAU scheme is combined with that of the Van Leer scheme. Furthermore, the hybrid dissipation term is only applied for the solution of momentum fluxes in numerical flux function. Based on the numerical test results, the hybrid scheme is deemed to be a successful improvement on the shock stability of SLAU scheme, without compromising on the efficiency and accuracy. Copyright © 2016 John Wiley & Sons, Ltd.     

A critical evaluation of seven discretization schemes for convection–diffusion equations

A comparative study of seven discretization schemes for the equations describing convection-diffusion transport phenomena is presented. The (differencing) schemes considered are the conventional central- and upwind-difference schemes, together with the Leonard,¹ Leonard upwind¹ and Leonard super upwind difference¹ schemes. Also tested are the so called locally exact difference scheme² and the quadratic-upstream difference scheme.^3,4 In multidimensional problems errors arise from ‘false-diffusion’ and function approximations. It is asserted that false diffusion is essentially a multidimensional source of error. No mesh constraints are associated with errors in function approximation and discretization. Hence errors associated with discretization only may be investigated via one-dimensional problems. Thus, although the above schemes have been tested for one- and two-dimensional flows with sources, only the former are presented here. For 1D flows, the Leonard super upwind difference scheme and the locally exact scheme are shown to be far superior in accuracy to the others at all Peclet numbers and for most source distributions, for the test cases considered. Furthermore, the latter is shown to be considerably cheaper in computational terms than the former. The stability of the schemes and their CPU time requirements are also discussed.     

A bore-capturing finite volume method for open-channel flows

A high-resolution finite volume hydrodynamic solver is presented for open-channel flows based on the 2D shallow water equations. This Godunov-type upwind scheme uses an efficient Harten–Lax–van Leer (HLL) approximate Riemann solver capable of capturing bore waves and simulating supercritical flows. Second-order accuracy is achieved by means of MUSCL reconstruction in conjunction with a Hancock two-stage scheme for the time integration. By using a finite volume approach, the computational grid can be irregular which allows for easy boundary fitting. The method can be applied directly to model 1D flows in an open channel with a rectangular cross-section without the need to modify the scheme. Such a modification is normally required for solving the 1D St Venant equations to take account of the variation of channel width. The numerical scheme and results of three test problems are presented in this paper. © 1998 John Wiley & Sons, Ltd.     

Verification of Euler/Navier–Stokes codes using the method of manufactured solutions

The method of manufactured solutions is used to verify the order of accuracy of two finite‐volume Euler and Navier–Stokes codes. The Premo code employs a node‐centred approach using unstructured meshes, while the Wind code employs a similar scheme on structured meshes. Both codes use Roe's upwind method with MUSCL extrapolation for the convective terms and central differences for the diffusion terms, thus yielding a numerical scheme that is formally second‐order accurate. The method of manufactured solutions is employed to generate exact solutions to the governing Euler and Navier–Stokes equations in two dimensions along with additional source terms. These exact solutions are then used to accurately evaluate the discretization error in the numerical solutions. Through global discretization error analyses, the spatial order of accuracy is observed to be second order for both codes, thus giving a high degree of confidence that the two codes are free from coding mistakes in the options exercised. Examples of coding mistakes discovered using the method are also given. Copyright © 2004 John Wiley & Sons, Ltd.     

Efficient construction of high‐resolution TVD conservative schemes for equations with source terms: application to shallow water flows

High‐resolution total variation diminishing (TVD) schemes are widely used for the numerical approximation of hyperbolic conservation laws. Their extension to equations with source terms involving spatial derivatives is not obvious. In this work, efficient ways of constructing conservative schemes from the conservative, non‐conservative or characteristic form of the equations are described in detail. An upwind, as opposed to a pointwise, treatment of the source terms is adopted here, and a new technique is proposed in which source terms are included in the flux limiter functions to get a complete second‐order compact scheme. A new correction to fix the entropy problem is also presented and a robust treatment of the boundary conditions according to the discretization used is stated. Copyright © 2001 John Wiley & Sons, Ltd.     

Insights on a sign‐preserving numerical method for the advection–diffusion equation

In this paper we explore theoretically and numerically the application of the advection transport algorithm introduced by Smolarkiewicz to the one‐dimensional unsteady advection–diffusion equation. The scheme consists of a sequence of upwind iterations, where the initial iteration is the first‐order accurate upwind scheme, while the subsequent iterations are designed to compensate for the truncation error of the preceding step. Two versions of the method are discussed. One, the classical version of the method, regards the second‐order terms of the truncation error and the other considers additionally the third‐order terms. Stability and convergence are discussed and the theoretical considerations are illustrated through numerical tests. The numerical tests will also indicate in which situations are advantageous to use the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Developing a physical influence upwind scheme for pressure-based cell-centered finite volume methods

In the framework of a cell-centered finite volume method (FVM), the advection scheme plays the most important role in developing FVMs to solve complicated fluid flow problems for a wide range of Reynolds numbers. Advection schemes have been widely developed for FVMs employing pressure-velocity coupling methodology in the incompressible flow limit. In this regard, the physical influence upwind scheme (PIS) is developed for a cell-centered finite volume coupled solver (FVCS) using a pressure-weighted interpolation method for linking the pressure and velocity fields. The well-known exponential differencing scheme and skew upwind differencing scheme are also deployed in the current FVCS and their numerical results are presented. The accuracy and convergence of the present PIS are evaluated solving flow in a lid-driven square cavity, a lid-driven skewed cavity, and over a backward-facing step (BFS). The flow within the lid-driven square cavity is numerically solved at Reynolds numbers from 400 to 10 000 on a relatively coarse mesh with respect to other reported solutions. The lid-driven skewed cavity test case at Reynolds number of 1000 demonstrates the numerical performance of the present PIS on nonorthogonal grids. The flow over a BFS at Reynolds number of 800 is numerically solved to examine capabilities of current FVCS employing the current PIS in inlet-outlet flow computations. The numerical results obtained by the current PIS are in excellent agreement with those of benchmark solutions of corresponding test cases. Incorporating implicit role of pressure terms in a pressure-weighted interpolation method and development of PIS provides satisfactory solution convergence alongside the numerical accuracy for the current FVCS. A particular numerical verification is performed for the V velocity calculation within the BFS flow field, which confirms the reliability of present PIS.     

SOLUTION OF THE ADVECTION–DIFFUSION EQUATION USING A COMBINATION OF DISCONTINUOUS AND MIXED FINITE ELEMENTS

When transport is advection-dominated, classical numerical methods introduce excessive artificial diffusion and spurious oscillations. Special methods are required to overcome these phenomena. To solve the advection‒diffusion equation, a numerical method is developed using a discontinuous finite element method for the discretization of the advective terms. At the discontinuities of the approximate solution, numerical advective fluxes are calculated using one-dimensional approximate Riemann solvers. The method is stabilized with a multidimensional slope limiter which introduces small amounts of numerical diffusion when sharp concentration fronts occur. In addition, the diffusive term is discretized using a mixed hybrid finite element method. With this approach, numerical oscillations are completely avoided for a full range of cell Peclet numbers. The combination of discontinuous and mixed finite elements can be easily applied to 2D and 3D models using various types of elements in regular and irregular meshes. Numerical tests show good agreement with 1D and 2D analytical solutions. This approach is compared at the same time with two different numerical methods, a standard mixed finite method and a finite volume approach with high-resolution upwind terms. Regular and irregular meshes are used for the numerical tests to study the mesh effects on the numerical results. Our data show that in all cases this approach performs well. © 1997 by John Wiley & Sons, Ltd.     

The influence of source terms on stability,accuracy and conservation in two‐dimensional shallow flow simulation using triangular finite volumes

The two‐dimensional shallow water model is a hyperbolic system of equations considered well suited to simulate unsteady phenomena related to some surface wave propagation. The development of numerical schemes to correctly solve that system of equations finds naturally an initial step in two‐dimensional scalar equation, homogeneous or with source terms. We shall first provide a complete formulation of the second‐order finite volume scheme for this equation, paying special attention to the reduction of the method to first order as a particular case. The explicit first and second order in space upwind finite volume schemes are analysed to provide an understanding of the stability constraints, making emphasis in the numerical conservation and in the preservation of the positivity property of the solution when necessary in the presence of source terms. The time step requirements for stability are defined at the cell edges, related with the traditional Courant–Friedrichs–Lewy (CFL) condition. Copyright © 2007 John Wiley & Sons, Ltd.     

The Osher scheme for non-equilibrium reacting flows

An extension of the Osher upwind scheme to non-equilibrium reacting flows is presented, Owing to the presence of source terms, the Riemann problem is no longer self-similar and therefore its approximate solution becomes tedious. With simplicity in mind, a linearized approach which avoids an iterative solution is used to define the intermediate states and sonic points. The source terms are treated explicitly. Numerical computations are presented to demonstrate the feasibility, efficiency and accuracy of the proposed method. The test problems include a ZND (Zeldovich-Neumann-Doring) detonation problem for which spurious numerical solutions which propagate at mesh speed have been observed on coarse grids. With the present method, a change of limiter causes the solution to change from the physically correct CJ detonation solution to the spurious weak detonation solution.