A nonlocal convective flux limiter for upwind‐biased finite volume simulations

Finite volume methods on structured and unstructured meshes commonly use second‐order, upwind‐biased linear reconstruction schemes to approximate the convective terms. Limiters are employed to reduce the inherent variable overshoot and undershoot of these schemes in discontinuous regions; however, they also can significantly increase the numerical dissipation of a solution in smooth regions. This paper presents a novel nonlocal, nonmonotonic (NLNM) limiter developed by enforcing cell minima and maxima on dependent variable values projected to cell faces. The minimum and maximum values for a cell are determined primarily through recursive reference to the minimum and maximum values of its upwind neighbors. The new limiter has been implemented into an existing flow solver. Various test cases are presented, which highlight the ability of the NLNM limiter to eliminate nonphysical oscillations while maintaining negligible levels of limiter‐induced dissipation into the solution. Results from the new limiter are compared with those from other limited and unlimited second‐order upwind and first‐order upwind schemes. For the cases examined in the study, the NLNM limiter was found to improve accuracy without significantly decreasing overall simulation efficiency and robustness.Copyright © 2011 John Wiley & Sons, Ltd.     

A high‐order discontinuous Galerkin method for all‐speed flows

In this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds‐averaged Navier–Stokes and k ? ω turbulence model equations for solving all‐speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non‐preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill‐conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean‐flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high‐lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high‐order DG solver at different flow regimes. Copyright © 2014 John Wiley & Sons, Ltd.     

A finite volume scheme preserving extremum principle for convection–diffusion equations on polygonal meshes

We propose a nonlinear finite volume scheme for convection–diffusion equation on polygonal meshes and prove that the discrete solution of the scheme satisfies the discrete extremum principle. The approximation of diffusive flux is based on an adaptive approach of choosing stencil in the construction of discrete normal flux, and the approximation of convection flux is based on the second‐order upwind method with proper slope limiter. Our scheme is locally conservative and has only cell‐centered unknowns. Numerical results show that our scheme can preserve discrete extremum principle and has almost second‐order accuracy. Copyright © 2017 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.     

Analysis of a discontinuous Galerkin approximation of the Stokes problem based on an artificial compressibility flux

In this work, we propose and analyse a discontinuous Galerkin (DG) method for the Stokes problem based on an artificial compressibility numerical flux. A crucial step in the definition of a DG method is the choice of the numerical fluxes, which affect both the accuracy and the order of convergence of the method. We propose here to treat the viscous and the inviscid terms separately. The former is discretized using the well‐known BRMPS method. For the latter, the problem is locally modified by adding an artificial compressibility term of the form (1/c²)(?p/?t) for the sole purpose of interface flux computation. The flux is obtained as the exact solution of a local Riemann problem. The analysis of the method extends the well‐established strategies for the DG discretization of the Laplacian to the resulting partially coercive problem. Copyright © 2007 John Wiley & Sons, Ltd.     

An improvement of classical slope limiters for high‐order discontinuous Galerkin method

In this paper, we describe some existing slope limiters (Cockburn and Shu's slope limiter and Hoteit's slope limiter) for the two‐dimensional Runge–Kutta discontinuous Galerkin (RKDG) method on arbitrary unstructured triangular grids. We describe the strategies for detecting discontinuities and for limiting spurious oscillations near such discontinuities, when solving hyperbolic systems of conservation laws by high‐order discontinuous Galerkin methods. The disadvantage of these slope limiters is that they depend on a positive constant, which is, for specific hydraulic problems, difficult to estimate in order to eliminate oscillations near discontinuities without decreasing the high‐order accuracy of the scheme in the smooth regions. We introduce the idea of a simple modification of Cockburn and Shu's slope limiter to avoid the use of this constant number. This modification consists in: slopes are limited so that the solution at the integration points is in the range spanned by the neighboring solution averages. Numerical results are presented for a nonlinear system: the shallow water equations. Four hydraulic problems of discontinuous solutions of two‐dimensional shallow water are presented. The idealized dam break problem, the oblique hydraulic jump problem, flow in a channel with concave bed and the dam break problem in a converging–diverging channel are solved by using the different slope limiters. Numerical comparisons on unstructured meshes show a superior accuracy with the modified slope limiter. Moreover, it does not require the choice of any constant number for the limiter condition. Copyright © 2008 John Wiley & Sons, Ltd.     

Hybrid finite‐volume finite‐difference scheme for the solution of Boussinesq equations

A hybrid scheme composed of finite‐volume and finite‐difference methods is introduced for the solution of the Boussinesq equations. While the finite‐volume method with a Riemann solver is applied to the conservative part of the equations, the higher‐order Boussinesq terms are discretized using the finite‐difference scheme. Fourth‐order accuracy in space for the finite‐volume solution is achieved using the MUSCL‐TVD scheme. Within this, four limiters have been tested, of which van‐Leer limiter is found to be the most suitable. The Adams–Basforth third‐order predictor and Adams–Moulton fourth‐order corrector methods are used to obtain fourth‐order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model ‘HYWAVE’, based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi‐chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd.     

Development of a class of multiple time‐stepping schemes for convection–diffusion equations in two dimensions

In this paper we present a class of semi‐discretization finite difference schemes for solving the transient convection–diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection–diffusion (CD) equation to the inhomogeneous steady convection–diffusion‐reaction (CDR) equation after using different time‐stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one‐dimensional framework. For the sake of increasing accuracy, the exact solution for the one‐dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one‐dimensional problem. Development of the proposed time‐stepping schemes is rooted in the Taylor series expansion. All higher‐order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection–diffusion‐reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

Time‐dependent finite element simulations of a shear‐thinning viscoelastic fluid with application to blood flow

A new finite element method is developed to simulate time‐dependent viscoelastic shear‐thinning flows characterized by the generalized Oldroyd‐B model. The focus of the algorithm is improved stability through a free‐energy dissipative scheme by using low‐order piecewise‐constant finite element approximations for stress. The algorithm is further modified by incorporating a pressure‐projection method, a DG‐upwinding scheme, a symmetric interior penalty DG method to solve the elliptic pressure‐update equation and a geometric multigrid preconditioner. The improved stability and cost to accuracy is compared when using higher order discontinuous bilinear approximation, where in addition, we consider the influence of a slope limiter for these elements. The algorithm is applied to the 2D start‐up‐driven cavity problem, and the stability of the free energy is illustrated and compared between element choices. An application of the model to modelling blood in small arterioles and channels is considered by simulating pulsatile blood flow through a stenotic arteriole. The individual influences of viscoelasticity and shear‐thinning within the generalized Oldroyd‐B model are investigated by comparing results to the Newtonian, generalized Newtonian and Oldroyd‐B models. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical solution of the incompressible Navier–Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier–Stokes equations is presented. The scheme is constructed on a staggered‐mesh grid system. The convection terms are discretized with a fifth‐order‐accurate upwind compact difference approximation, the viscous terms are discretized with a sixth‐order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth‐order difference approximation on a cell‐centered mesh. Time advancement uses a three‐stage Runge–Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

On the treatment of transient area variation in 1D discontinuous Galerkin simulations of train‐induced pressure waves in tunnels

To simulate the pressure wave generated by a train travelling through a tunnel, we implement a discontinuous Galerkin (DG) method for the solution of the one‐dimensional equations of variable area flow. This formulation uses a spatial discretisation via Legendre polynomials of arbitrary degree, and the resulting semi‐discrete system is integrated using an explicit Runge–Kutta scheme. A simulation of subsonic steady flow in a nozzle shows that the scheme produces stable solutions, without the need for artificial dissipation, and that its performance is optimal for polynomial degrees between 5 and 7. However, when dealing with an unsteady area, we report the presence of numerical oscillations that are not due to the steep pressure fronts in the flow but rather to the projection of a moving area, with piecewise continuous derivatives onto a fixed grid. We propose a reformulation of the DG method to eliminate these oscillations that, put in simple terms, amount to splitting the integrals where the derivatives of the cross‐sectional area are discontinuous into subintegrals where they are continuous. The resulting method does not exhibit oscillations, and it is applied here to two practical cases involving train‐induced pressure waves in a tunnel. The first application is a validation of the DG method through comparison of its computational results with pressure data measured during transit at the Patchway tunnel near Bristol (UK). The second application is a study of the influence of the nose shape and length on the pressure wave gradients responsible for sonic boom at tunnel exit portals to show that the proposed modification is able to deal with realistic train shapes. Copyright © 2012 John Wiley & Sons, Ltd.     

Coupling of finite element method and discontinuous Galerkin method to simulate viscoelastic flows

In this paper, a numerical method, which is about the coupling of continuous and discontinuous Galerkin method based on the splitting scheme, is presented for the calculation of viscoelastic flows of the Oldroyd‐B fluid. The momentum equation is discretized in time by using the Adams‐Bashforth second‐order algorithm, and then decoupled via the splitting approach. Considering the Oldroyd‐B constitutive equation, the second‐order Runge‐Kutta approach is selected to complete the temporal discretization. As for the spatial discretizations, the fundamental purpose is to make the best of finite element method (FEM) and discontinuous Galerkin (DG) method to handle different types of equations. Specifically speaking, for the subequations, FEM is chosen to treat the Poisson and Helmholtz equations, and DG is employed to deal with the nonlinear convective term. In addition, because of the hyperbolic nature, DG is also utilized to discretize the Oldroyd‐B constitutive equation spatially. This coupled method avoids resorting to extra stabilization technique occurred in standard FEM framework even for moderately high values of Weissenberg number and also reduces the complexity compared with unified DG scheme. The Oldroyd‐B model is applied to investigate several typical and challenging benchmarks, such as the 4:1 planar contraction flow and the lid‐driven cavity flow, with a wide range of Weissenberg number to illustrate the feasibility, robustness, and validity of our coupled method.     

Implicit FEM‐FCT algorithms and discrete Newton methods for transient convection problems

A new generalization of the flux‐corrected transport (FCT) methodology to implicit finite element discretizations is proposed. The underlying high‐order scheme is supposed to be unconditionally stable and produce time‐accurate solutions to evolutionary convection problems. Its nonoscillatory low‐order counterpart is constructed by means of mass lumping followed by elimination of negative off‐diagonal entries from the discrete transport operator. The raw antidiffusive fluxes, which represent the difference between the high‐ and low‐order schemes, are updated and limited within an outer fixed‐point iteration. The upper bound for the magnitude of each antidiffusive flux is evaluated using a single sweep of the multidimensional FCT limiter at the first outer iteration. This semi‐implicit limiting strategy makes it possible to enforce the positivity constraint in a very robust and efficient manner. Moreover, the computation of an intermediate low‐order solution can be avoided. The nonlinear algebraic systems are solved either by a standard defect correction scheme or by means of a discrete Newton approach, whereby the approximate Jacobian matrix is assembled edge by edge. Numerical examples are presented for two‐dimensional benchmark problems discretized by the standard Galerkin finite element method combined with the Crank–Nicolson time stepping. Copyright © 2007 John Wiley & Sons, Ltd.     

A high‐order alternating direction implicit method for the unsteady convection‐dominated diffusion problem

A high‐order alternating direction implicit (ADI) method for solving the unsteady convection‐dominated diffusion equation is developed. The fourth‐order Padé scheme is used for the discretization of the convection terms, while the second‐order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two‐dimensional problem becomes a sequence of one‐dimensional problems. The solution procedure consists of multiple use of a one‐dimensional tridiagonal matrix algorithm that produces a computationally cost‐effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two‐dimensional problem concerning convection‐dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high‐order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection‐dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high‐order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.     

A hybrid Padé ADI scheme of higher‐order for convection–diffusion problems

A high‐order Padé alternating direction implicit (ADI) scheme is proposed for solving unsteady convection–diffusion problems. The scheme employs standard high‐order Padé approximations for spatial first and second derivatives in the convection‐diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22 : 87–110) are applied for the time integration. The approximate factorization imposes a second‐order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher‐order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC‐based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high‐order ADI schemes for solving unsteady convection‐diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given. Copyright © 2009 John Wiley & Sons, Ltd.     

A high‐order upwind control‐volume method based on integrated RBFs for fluid‐flow problems

This paper is concerned with the development of a high‐order upwind conservative discretization method for the simulation of flows of a Newtonian fluid in two dimensions. The fluid‐flow domain is discretized using a Cartesian grid from which non‐overlapping rectangular control volumes are formed. Line integrals arising from the integration of the diffusion and convection terms over control volumes are evaluated using the middle‐point rule. One‐dimensional integrated radial basis function schemes using the multiquadric basis function are employed to represent the variations of the field variables along the grid lines. The convection term is effectively treated using an upwind scheme with the deferred‐correction strategy. Several highly non‐linear test problems governed by the Burgers and the Navier–Stokes equations are simulated, which show that the proposed technique is stable, accurate and converges well. Copyright © 2010 John Wiley & Sons, Ltd.     

A lattice Boltzmann model for simulation of compressible flows

This paper presents a new model of lattice Boltzmann method for full compressible flows. On the basis of multi‐speed model, an extra potential energy distribution function is introduced to recover the full compressible Navier–Stokes equations with a flexible specific‐heat ratio and Prandtl number. The Chapman–Enskog expansion of the kinetic equations is performed, and the two‐dimension‐seventeen‐velocity density equilibrium distribution functions are obtained. The governing equations are discretized using the third order monotone upwind scheme for scalar conservation laws finite volume scheme. The van Albada limiter is used to avoid spurious oscillations. In order to verify the accuracy of this double‐distribution‐function model, the Riemann problems, Couette flows, and flows around a NACA0012 airfoil are simulated. It is found that the proposed lattice Boltzmann model is suitable for compressible flows, even for strong shock wave problem, which has an extremely large pressure ratio, 100,000. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

基于静动态混合重构的DG/FV混合格式

通过比较紧致格式和间断Galerkin(DG)格式, 提出了``静态重构'和``动态重构'的概念,对有限体积方法和DG有限元方法进行统一的表述. 借鉴有限体积的思想, 发展了基于``混合重构'技术的一类新的DG格式, 称之为间断Galerkin有限元/有限体积混合格式(DG/FV格式). 该类混合格式通过适当地扩展模板(拓展至紧邻单元)重构单元内的高阶多项式分布, 在提高精度的同时, 减少了传统DG格式的计算量和存储量. 通过典型一维和二维标量方程的计算发现新的混合格式在有些情况下具有超收敛(superconvergence)性质.      

A simple shock‐capturing technique for high‐order discontinuous Galerkin methods

This article presents a novel shock‐capturing technique for the discontinuous Galerkin (DG) method. The technique is designed for compressible flow problems, which are usually characterized by the presence of strong shocks and discontinuities. The inherent structure of standard DG methods seems to suggest that they are especially adapted to capture shocks because of the numerical fluxes based on suitable approximate Riemann solvers, which, in practice, introduces some stabilization. However, the usual numerical fluxes are not sufficient to stabilize the solution in the presence of shocks for large high‐order elements. Here, a new basis of shape functions is introduced. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization because of numerical fluxes. Large high‐order elements can therefore be used and shocks captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Several numerical examples for transonic and supersonic flows are studied to demonstrate the applicability of the proposed approach. Copyright © 2011 John Wiley & Sons, Ltd.