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.     

An adaptive cut‐cell method for environmental fluid mechanics

In this work we present a numerical method for solving the incompressible Navier–Stokes equations in an environmental fluid mechanics context. The method is designed for the study of environmental flows that are multiscale, incompressible, variable‐density, and within arbitrarily complex and possibly anisotropic domains. The method is new because in this context we couple the embedded‐boundary (or cut‐cell) method for complex geometry with block‐structured adaptive mesh refinement (AMR) while maintaining conservation and second‐order accuracy. The accurate simulation of variable‐density fluids necessitates special care in formulating projection methods. This variable‐density formulation is well known for incompressible flows in unit‐aspect ratio domains, without AMR, and without complex geometry, but here we carefully present a new method that addresses the intersection of these issues. The methodology is based on a second‐order‐accurate projection method with high‐order‐accurate Godunov finite‐differencing, including slope limiting and a stable differencing of the nonlinear convection terms. The finite‐volume AMR discretizations are based on two‐way flux matching at refinement boundaries to obtain a conservative method that is second‐order accurate in solution error. The control volumes are formed by the intersection of the irregular embedded boundary with Cartesian grid cells. Unlike typical discretization methods, these control volumes naturally fit within parallelizable, disjoint‐block data structures, and permit dynamic AMR coarsening and refinement as the simulation progresses. We present two‐ and three‐dimensional numerical examples to illustrate the accuracy of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Refining the submesh strategy in the two‐level finite element method: application to the advection–diffusion equation

We introduce a new submesh strategy for the two‐level finite element method. The numerical results show that the new submesh is able to better capture the boundary layer which is caused by the choice of bubble functions. The effect of an improved approximation of the residual free bubbles is studied for the advective–diffusive equation. Copyright © 2002 John Wiley & Sons, Ltd.     

An upstream flux‐splitting finite‐volume scheme for 2D shallow water equations

An upstream flux‐splitting finite‐volume (UFF) scheme is proposed for the solutions of the 2D shallow water equations. In the framework of the finite‐volume method, the artificially upstream flux vector splitting method is employed to establish the numerical flux function for the local Riemann problem. Based on this algorithm, an UFF scheme without Jacobian matrix operation is developed. The proposed scheme satisfying entropy condition is extended to be second‐order‐accurate using the MUSCL approach. The proposed UFF scheme and its second‐order extension are verified through the simulations of four shallow water problems, including the 1D idealized dam breaking, the oblique hydraulic jump, the circular dam breaking, and the dam‐break experiment with 45° bend channel. Meanwhile, the numerical performance of the UFF scheme is compared with those of three well‐known upwind schemes, namely the Osher, Roe, and HLL schemes. It is demonstrated that the proposed scheme performs remarkably well for shallow water flows. The simulated results also show that the UFF scheme has superior overall numerical performances among the schemes tested. Copyright © 2005 John Wiley & Sons, Ltd.     

A physically based flux limiter for QUICK calculations of advective scalar transport

Transient, advective transport of a contaminant into a clean domain will exhibit a moving sharp front that separates contaminated and clean regions. Due to ‘numerical diffusion’—the combined effects of ‘cross‐wind diffusion’ and ‘artificial dispersion’—a numerical solution based on a first‐order (upwind) treatment will smear out the sharp front. The use of higher‐order schemes, e.g. QUICK (quadratic upwinding) reduces the smearing but can introduce non‐physical oscillations in the solution. A common approach to reduce numerical diffusion without oscillations is to use a scheme that blends low‐order and high‐order approximations of the advective transport. Typically, the blending is based on a parameter that measures the local monotonicity in the predicted scalar field. In this paper, an alternative approach is proposed for use in scalar transport problems where physical bounds C_Low?C?C_High on the scalar are known a priori. For this class of problems, the proposed scheme switches from a QUICK approximation to an upwind approximation whenever the predicted upwind nodal value falls outside of the physical range [C_Low, C_High]. On two‐dimensional steady‐state and one‐dimensional transient test problems predictions obtained with the proposed scheme are essentially indistinguishable from those obtained with monotonic flux‐limiter schemes. An analysis of the modified equation explains the observed performance of first‐ and second‐order time‐stepping schemes in predicting the advective transport of a step. In application to the transient two‐dimensional problem of contaminate transport into a streambed, predictions obtained with the proposed flux‐limiter scheme agree with those obtained with a scheme from the literature. Copyright © 2007 John Wiley & Sons, Ltd.     

Effects of the Jacobian evaluation on Newton's solution of the Euler equations

Newton's method is developed for solving the 2‐D Euler equations. The Euler equations are discretized using a finite‐volume method with upwind flux splitting schemes. Both analytical and numerical methods are used for Jacobian calculations. Although the numerical method has the advantage of keeping the Jacobian consistent with the numerical residual vector and avoiding extremely complex analytical differentiations, it may have accuracy problems and need longer execution time. In order to improve the accuracy of numerical Jacobians, detailed error analyses are performed. Results show that the finite‐difference perturbation magnitude and computer precision are the most important parameters that affect the accuracy of numerical Jacobians. A method is developed for calculating an optimal perturbation magnitude that can minimize the error in numerical Jacobians. The accuracy of the numerical Jacobians is improved significantly by using the optimal perturbation magnitude. The effects of the accuracy of numerical Jacobians on the convergence of the flow solver are also investigated. In order to reduce the execution time for numerical Jacobian evaluation, flux vectors with perturbed flow variables are calculated only for neighbouring cells. A sparse matrix solver that is based on LU factorization is used. Effects of different flux splitting methods and higher‐order discretizations on the performance of the solver are analysed. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

A discontinuous Galerkin method for the Navier–Stokes equations

The foundations of a new discontinuous Galerkin method for simulating compressible viscous flows with shocks on standard unstructured grids are presented in this paper. The new method is based on a discontinuous Galerkin formulation both for the advective and the diffusive contributions. High‐order accuracy is achieved by using a recently developed hierarchical spectral basis. This basis is formed by combining Jacobi polynomials of high‐order weights written in a new co‐ordinate system. It retains a tensor‐product property, and provides accurate numerical quadrature. The formulation is conservative, and monotonicity is enforced by appropriately lowering the basis order and performing h‐refinement around discontinuities. Convergence results are shown for analytical two‐ and three‐dimensional solutions of diffusion and Navier–Stokes equations that demonstrate exponential convergence of the new method, even for highly distorted elements. Flow simulations for subsonic, transonic and supersonic flows are also presented that demonstrate discretization flexibility using hp‐type refinement. Unlike other high‐order methods, the new method uses standard finite volume grids consisting of arbitrary triangulizations and tetrahedrizations. Copyright © 1999 John Wiley & Sons, Ltd.     

Monotone nonlinear finite‐volume method for nonisothermal two‐phase two‐component flow in porous media

This article presents a new nonlinear finite‐volume scheme for the nonisothermal two‐phase two‐component flow equations in porous media. The face fluxes are approximated by a nonlinear two‐point flux approximation, where transmissibilities nonlinearly depend on primary variables. Thereby, we mainly follow the ideas proposed by Le Potier combined with a harmonic averaging point interpolation strategy for the approximation of arbitrary heterogeneous permeability fields on polygonal grids. The behavior of this interpolation strategy is analyzed, and its limitation for highly anisotropic permeability tensors is demonstrated. Moreover, the condition numbers of occurring matrices are compared with linear finite‐volume schemes. Additionally, the convergence behavior of iterative solvers is investigated. Finally, it is shown that the nonlinear scheme is more efficient than its linear counterpart. Copyright © 2016 John Wiley & Sons, Ltd.     

A numerical method for flows in porous and homogenous fluid domains coupled at the interface by stress jump

A numerical method was developed for flows involving an interface between a homogenous fluid and a porous medium. The numerical method is based on the finite volume method with body‐fitted and multi‐block grids. A generalized model, which includes Brinkman term, Forcheimmer term and non‐linear convective term, was used to govern the flow in the porous medium region. At its interface, a shear stress jump that includes the inertial effect was imposed, together with a continuity of normal stress. Furthermore, the effect of the jump condition on the diffusive flux was considered, additional to that on the convective part which has been usually considered. Numerical results of three flow configurations are presented. The method is suitable for coupled problems with regions of homogeneous fluid and porous medium, which have complex geometries. Copyright © 2006 John Wiley & Sons, Ltd.     

Multidimensional FEM‐FCT schemes for arbitrary time stepping

The flux‐corrected‐transport paradigm is generalized to finite‐element schemes based on arbitrary time stepping. A conservative flux decomposition procedure is proposed for both convective and diffusive terms. Mathematical properties of positivity‐preserving schemes are reviewed. A nonoscillatory low‐order method is constructed by elimination of negative off‐diagonal entries of the discrete transport operator. The linearization of source terms and extension to hyperbolic systems are discussed. Zalesak's multidimensional limiter is employed to switch between linear discretizations of high and low order. A rigorous proof of positivity is provided. The treatment of non‐linearities and iterative solution of linear systems are addressed. The performance of the new algorithm is illustrated by numerical examples for the shock tube problem in one dimension and scalar transport equations in two dimensions. Copyright © 2003 John Wiley & Sons, Ltd.     

A CIP/multi‐moment finite volume method for shallow water equations with source terms

A novel finite volume method has been presented to solve the shallow water equations. In addition to the volume‐integrated average (VIA) for each mesh cell, the surface‐integrated average (SIA) is also treated as the model variable and is independently predicted. The numerical reconstruction is conducted based on both the VIA and the SIA. Different approaches are used to update VIA and SIA separately. The SIA is updated by a semi‐Lagrangian scheme in terms of the Riemann invariants of the shallow water equations, while the VIA is computed by a flux‐based finite volume formulation and is thus exactly conserved. Numerical oscillation can be effectively avoided through the use of a non‐oscillatory interpolation function. The numerical formulations for both SIA and VIA moments maintain exactly the balance between the fluxes and the source terms. 1D and 2D numerical formulations are validated with numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.     

A FE‐based algorithm for the inverse natural convection problem

Several numerical algorithms for solving inverse natural convection problems are revisited and studied. Our aim is to identify the unknown strength of a time‐varying heat source via a set of coupled nonlinear partial differential equations obtained by the so‐called finite element consistent splitting scheme (CSS) in order to get a good approximation of the unknown heat source from both the measured data and model results, by minimizing a functional that measures discrepancies between model and measured data. Viewed as an optimization problem, the solutions are obtained by means of the conjugate gradient method. A second‐order CSS in time involving the direct problem, the adjoint problem, the sensitivity problem and a system of sensitivity functions is used in order to enhance the numerical accuracy obtained for the unknown heat source function. A spatial discretization of all field equations is implemented using equal‐order and mixed finite element methods. Numerical experiments validate the proposed optimization algorithms that are in good agreement with the existing results. Copyright © 2010 John Wiley & Sons, Ltd.     

Geometrical interpretation of the multi‐point flux approximation L‐method

In this paper, we first investigate the influence of different Dirichlet boundary discretizations on the convergence rate of the multi‐point flux approximation (MPFA) L‐method by the numerical comparisons between the MPFA O‐ and L‐method, and show how important it is for this new method to handle Dirichlet boundary conditions in a suitable way. A new Dirichlet boundary strategy is proposed, which in some sense can well recover the superconvergence rate of the normal velocity. In the second part of the work, the MPFA L‐method with homogeneous media is studied. A systematic concept and geometrical interpretations of the L‐method are given and illustrated, which yield more insight into the L‐method. Finally, we apply the MPFA L‐method for two‐phase flow in porous media on different quadrilateral grids and compare its numerical results for the pressure and saturation with the results of the two‐point flux approximation method. Copyright © 2008 John Wiley & Sons, Ltd.     

Velocity–pressure coupling in finite difference formulations for the Navier–Stokes equations

A new numerical procedure for solving the two‐dimensional, steady, incompressible, viscous flow equations on a staggered Cartesian grid is presented in this paper. The proposed methodology is finite difference based, but essentially takes advantage of the best features of two well‐established numerical formulations, the finite difference and finite volume methods. Some weaknesses of the finite difference approach are removed by exploiting the strengths of the finite volume method. In particular, the issue of velocity–pressure coupling is dealt with in the proposed finite difference formulation by developing a pressure correction equation using the SIMPLE approach commonly used in finite volume formulations. However, since this is purely a finite difference formulation, numerical approximation of fluxes is not required. Results presented in this paper are based on first‐ and second‐order upwind schemes for the convective terms. This new formulation is validated against experimental and other numerical data for well‐known benchmark problems, namely developing laminar flow in a straight duct, flow over a backward‐facing step, and lid‐driven cavity flow. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite volume solution of the Navier–Stokes equations in velocity–vorticity formulation

For the incompressible Navier–Stokes equations, vorticity‐based formulations have many attractive features over primitive‐variable velocity–pressure formulations. However, some features interfere with the use of the numerical methods based on the vorticity formulations, one of them being the lack of a boundary conditions on vorticity. In this paper, a novel approach is presented to solve the velocity–vorticity integro‐differential formulations. The general numerical method is based on standard finite volume scheme. The velocities needed at the vertexes of each control volume are calculated by a so‐called generalized Biot–Savart formula combined with a fast summation algorithm, which makes the velocity boundary conditions implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields. The well‐known fractional step approaches are used to solve the vorticity transport equation. The paper describes in detail how we accurately impose no normal‐flow and no tangential‐flow boundary conditions. We impose a no‐flux boundary condition on solid objects by the introduction of a proper amount of vorticity at wall. The diffusion term in the transport equation is treated implicitly using a conservative finite update. The diffusive fluxes of vorticity into flow domain from solid boundaries are determined by an iterative process in order to satisfy the no tangential‐flow boundary condition. As application examples, the impulsively started flows through a flat plate and a circular cylinder are computed using the method. The present results are compared with the analytical solution and other numerical results and show good agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid

This paper presents a finite element method for incompressible multiphase flows with capillary interfaces based on a (formally) second‐order projection scheme. The discretization is on a fixed Eulerian grid. The fluid phases are identified and advected using a level set function. The grid is temporarily adapted around the interfaces in order to maintain optimal interpolations accounting for the pressure jump and the discontinuity of the normal velocity derivatives. The least‐squares method for computing the curvature is used, combined with piecewise linear approximation to the interface. The time integration is based on a formally second order splitting scheme. The convection substep is integrated over an Eulerian grid using an explicit scheme. The remaining generalized Stokes problem is solved by means of a formally second order pressure‐stabilized projection scheme. The pressure boundary condition on the free interface is imposed in a strong form (pointwise) at the pressure‐computation substep. This allows capturing significant pressure jumps across the interface without creating spurious instabilities. This method is simple and efficient, as demonstrated by the numerical experiments on a wide range of free‐surface problems. Copyright © 2004 John Wiley & Sons, Ltd.     

A time‐implicit high‐order compact differencing and filtering scheme for large‐eddy simulation

This work investigates a high‐order numerical method which is suitable for performing large‐eddy simulations, particularly those containing wall‐bounded regions which are considered on stretched curvilinear meshes. Spatial derivatives are represented by a sixth‐order compact approximation that is used in conjunction with a tenth‐order non‐dispersive filter. The scheme employs a time‐implicit approximately factored finite‐difference algorithm, and applies Newton‐like subiterations to achieve second‐order temporal and sixth‐order spatial accuracy. Both the Smagorinsky and dynamic subgrid‐scale stress models are incorporated in the computations, and are used for comparison along with simulations where no model is employed. Details of the method are summarized, and a series of classic validating computations are performed. These include the decay of compressible isotropic turbulence, turbulent channel flow, and the subsonic flow past a circular cylinder. For each of these cases, it was found that the method was robust and provided an accurate means of describing the flowfield, based upon comparisons with previous existing numerical results and experimental data. Published in 2003 by John Wiley & Sons, Ltd.     

Time‐accurate stabilized finite‐element model for weakly nonlinear and weakly dispersive water waves

Introduction of a time‐accurate stabilized finite‐element approximation for the numerical investigation of weakly nonlinear and weakly dispersive water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the linear triangular elements by the Galerkin finite‐element method, the fourth‐order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The streamline‐upwind Petrov–Galerkin (SUPG) method with crosswind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection‐dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Treatments of various boundary conditions, including the open boundary conditions, the perfect reflecting boundary conditions along boundaries with irregular geometry, are also described. Numerical results showing the comparisons with analytical solutions, experimental measurements, and other published numerical results are presented and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

A deconvolution‐based fourth‐order finite volume method for incompressible flows on non‐uniform grids

This paper is concerned with the development of a new high‐order finite volume method for the numerical simulation of highly convective unsteady incompressible flows on non‐uniform grids. Specifically, both a high‐order fluxes integration and the implicit deconvolution of the volume‐averaged field are considered. This way, the numerical solution effectively stands for a fourth‐order approximation of the point‐wise one. Moreover, the procedure is developed in the framework of a projection method for the pressure–velocity decoupling, while originally deriving proper high‐order intermediate boundary conditions. The entire numerical procedure is discussed in detail, giving particular attention to the consistent discretization of the deconvolution operation. The present method is also cast in the framework of approximate deconvolution modelling for large‐eddy simulation. The overall high accuracy of the method, both in time and space, is demonstrated. Finally, as a model of real flow computation, a two‐dimensional time‐evolving mixing layer is simulated, with and without sub‐grid scales modelling. Copyright © 2003 John Wiley & Sons, Ltd.