A family of multi‐point flux approximation schemes for general element types in two and three dimensions with convergence performance

A family of flux‐continuous, locally conservative, control‐volume‐distributed multi‐point flux approximation (CVD‐MPFA) schemes has been developed for solving the general geometry‐permeability tensor pressure equation on structured and unstructured grids. These schemes are applicable to the full‐tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full‐tensor flow approximation. The family of flux‐continuous schemes is characterized by a quadrature parameterization. Improved numerical convergence for the family of CVD‐MPFA schemes using the quadrature parameterization has been observed for structured and unstructured grids in two dimensions. The CVD‐MPFA family cell‐vertex formulation is extended to classical general element types in 3‐D including prisms, pyramids, hexahedra and tetrahedra. A numerical convergence study of the CVD‐MPFA schemes on general unstructured grids comprising of triangular elements in 2‐D and prismatic, pyramidal, hexahedral and tetrahedral shape elements in 3‐D is presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Convergence of MPFA on triangulations and for Richards' equation

Spatial discretization of transport and transformation processes in porous media requires techniques that handle general geometry, discontinuous coefficients and are locally mass conservative. Multi‐point flux approximation (MPFA) methods are such techniques, and we will here discuss some formulations on triangular grids with further application to the nonlinear Richards equation. The MPFA methods will be rewritten to mixed form to derive stability conditions and error estimates. Several MPFA versions will be shown, and the versions will be discussed with respect to convergence, symmetry and robustness when the grids are rough. It will be shown that the behavior may be quite different for challenging cases of skewness and roughness of the simulation grids. Further, we apply the MPFA discretization approach for the Richards equation and derive new error estimates without extra regularity requirements. The analysis will be accompanied by numerical results for grids that are relevant for practical simulation. Copyright © 2008 John Wiley & Sons, Ltd.     

A monotone finite volume scheme for advection–diffusion equations on distorted meshes

A new monotone finite volume method with second‐order accuracy is presented for the steady‐state advection–diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes that guarantee the positivity of the numerical solution. The approximation of the diffusive flux is based on nonlinear two‐point approximation, and the approximation of the advective flux is based on the second‐order upwind method with proper slope limiter. The second‐order convergence rate for concentration and the monotonicity of the nonlinear finite volume method are verified with numerical experiments. Copyright © 2011 John Wiley & Sons, Ltd.     

Anisotropy favoring triangulation CVD(MPFA) finite‐volume approximations

A family of flux‐continuous, control‐volume distributed multi‐point flux approximation schemes CVD (MPFA) have been developed for solving the general geometry‐permeability tensor pressure equation on structured and unstructured grids (Comput. Geo. 1998; 2 : 259–290, Comput. Geo. 2002; 6 : 433–452). The locally conservative schemes are applicable to the diagonal and full‐tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full‐tensor flow approximation. The family of flux‐continuous schemes is quantified by a quadrature parameterization. Improved numerical convergence for the family of CVD(MPFA) schemes for specified quadrature points has been observed for lower anisotropy ratios for both structured and unstructured grids in two dimensions. However, for strong full‐tensor anisotropy fields the quadrilateral schemes can induce strong spurious oscillations in the numerical solution. This paper motivates and demonstrates the benefit of using anisotropy favoring triangulation for treating such cases. Test examples involving strong full‐tensor anisotropy fields are presented in 2‐D and 3‐D, which show that the family of schemes on anisotropy favoring triangulation (prisms in 3‐D) yield well‐resolved pressure fields with little or no spurious oscillations. Copyright © 2010 John Wiley & Sons, Ltd.     

The effects of control‐volume distributed multi‐point flux approximation (CVD‐MPFA) on upscaling—A study using the CVD‐MPFA schemes

A family of flux‐continuous, locally conservative, finite‐volume schemes has been developed for solving the general geometry‐permeability tensor (petroleum reservoir‐simulation) pressure equation on structured and unstructured grids and are control‐volume distributed (textit Comput. Geo. 1998; 2 :259–290; Comput. Geo. 2002; 6 :433–452). The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir‐simulation schemes (two‐point flux approximation) when applied to full tensor flow approximation. The family of flux‐continuous schemes is quantified by a quadrature parameterization (Int. J. Numer. Meth. Fluids 2006; 51 :1177–1203). Improved convergence (for two‐ and three‐dimensional formulation) using the quadrature parameterization has been observed for the family of flux‐continuous control‐volume distributed multi‐point flux approximation (CVD‐MPFA) schemes (Ph.D. Thesis, University of Wales, Swansea, U.K., 2007). In this paper family of flux‐continuous (CVD‐MPFA) schemes are used as a part of numerical upscaling procedure for upscaling the fine‐scale grid information (permeability) onto a coarse grid scale. A series of data‐sets (SPE, 2001) are tested where the upscaled permeability tensor is computed on a sequence of grid levels using the same fixed range of quadrature points in each case. The refinement studies presented involve:

• (i) Refinement comparison study: In this study, permeability distribution for cells at each grid level is obtained by upscaling directly from the fine‐scale permeability field as in standard simulation practice.
• (ii) Refinement study with renormalized permeability: In this refinement comparison, the local permeability is upscaled to the next grid level hierarchically, so that permeability values are renormalized to each coarser level. Hence, showing only the effect of increased grid resolution on upscaled permeability, compared with that obtained directly from the fine‐scale solution.
• (iii) Refinement study with invariant permeability distribution: In this study, a classical mathematical convergence test is performed. The same coarse‐scale underlying permeability map is preserved on all grid levels including the fine‐scale reference solution.

The study is carried out for the discretization of the scheme in physical space. The benefit of using specific quadrature points is demonstrated for upscaling in this study and superconvergence is observed. Copyright © 2010 John Wiley & Sons, Ltd.     

Vortex particle‐in‐cell method for three‐dimensional viscous unbounded flow computations

A new vortex particle‐in‐cell (PIC) method is developed for the computation of three‐dimensional unsteady, incompressible viscous flow in an unbounded domain. The method combines the advantages of the Lagrangian particle methods for convection and the use of an Eulerian grid to compute the diffusion and vortex stretching. The velocity boundary conditions used in the method are of Dirichlet‐type, and can be calculated using the vorticity field on the grid by the Biot–Savart equation. The present results for the propagation speed of the single vortex ring are in good agreement with the Saffman's model. The applications of the method to the head‐on and head‐off collisions of the two vortex rings show good agreement with the experimental and numerical literature. Copyright © 2000 John Wiley & Sons, Ltd.     

Positive‐definite q‐families of continuous subcell Darcy‐flux CVD(MPFA) finite‐volume schemes and the mixed finite element method

A new family of locally conservative cell‐centred flux‐continuous schemes is presented for solving the porous media general‐tensor pressure equation. A general geometry‐permeability tensor approximation is introduced that is piecewise constant over the subcells of the control volumes and ensures that the local discrete general tensor is elliptic. A family of control‐volume distributed subcell flux‐continuous schemes are defined in terms of the quadrature parametrization q (Multigrid Methods. Birkhauser: Basel, 1993; Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, June 1994; Comput. Geosci. 1998; 2 :259–290), where the local position of flux continuity defines the quadrature point and each particular scheme. The subcell tensor approximation ensures that a symmetric positive‐definite (SPD) discretization matrix is obtained for the base member (q=1) of the formulation. The physical‐space schemes are shown to be non‐symmetric for general quadrilateral cells. Conditions for discrete ellipticity of the non‐symmetric schemes are derived with respect to the local symmetric part of the tensor. The relationship with the mixed finite element method is given for both the physical‐space and subcell‐space q‐families of schemes. M‐matrix monotonicity conditions for these schemes are summarized. A numerical convergence study of the schemes shows that while the physical‐space schemes are the most accurate, the subcell tensor approximation reduces solution errors when compared with earlier cell‐wise constant tensor schemes and that subcell tensor approximation using the control‐volume face geometry yields the best SPD scheme results. A particular quadrature point is found to improve numerical convergence of the subcell schemes for the cases tested. Copyright © 2007 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 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.     

A subdomain boundary element method for high‐Reynolds laminar flow using stream function‐vorticity formulation

The paper presents a new formulation of the integral boundary element method (BEM) using subdomain technique. A continuous approximation of the function and the function derivative in the direction normal to the boundary element (further ‘normal flux’) is introduced for solving the general form of a parabolic diffusion‐convective equation. Double nodes for normal flux approximation are used. The gradient continuity is required at the interior subdomain corners where compatibility and equilibrium interface conditions are prescribed. The obtained system matrix with more equations than unknowns is solved using the fast iterative linear least squares based solver. The robustness and stability of the developed formulation is shown on the cases of a backward‐facing step flow and a square‐driven cavity flow up to the Reynolds number value 50 000. Copyright © 2004 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.     

Constrained moving least‐squares immersed boundary method for fluid‐structure interaction analysis

A numerical method is presented for the analysis of interactions of inviscid and compressible flows with arbitrarily shaped stationary or moving rigid solids. The fluid equations are solved on a fixed rectangular Cartesian grid by using a higher‐order finite difference method based on the fifth‐order WENO scheme. A constrained moving least‐squares sharp interface method is proposed to enforce the Neumann‐type boundary conditions on the fluid‐solid interface by using a penalty term, while the Dirichlet boundary conditions are directly enforced. The solution of the fluid flow and the solid motion equations is advanced in time by staggerly using, respectively, the third‐order Runge‐Kutta and the implicit Newmark integration schemes. The stability and the robustness of the proposed method have been demonstrated by analyzing 5 challenging problems. For these problems, the numerical results have been found to agree well with their analytical and numerical solutions available in the literature. Effects of the support domain size and values assigned to the penalty parameter on the stability and the accuracy of the present method are also discussed.     

A finite volume method for numerical grid generation

A novel method to generate body‐fitted grids based on the direct solution for three scalar functions is derived. The solution for scalar variables ξ, η and ζ is obtained with a conventional finite volume method based on a physical space formulation. The grid is adapted or re‐zoned to eliminate the residual error between the current solution and the desired solution, by means of an implicit grid‐correction procedure. The scalar variables are re‐mapped and the process is reiterated until convergence is obtained. Calculations are performed for a variety of problems by assuming combined Dirichlet–Neumann and pure Dirichlet boundary conditions involving the use of transcendental control functions, as well as functions designed to effect grid control automatically on the basis of boundary values. The use of dimensional analysis to build stable exponential functions and other control functions is demonstrated. Automatic procedures are implemented: one based on a finite difference approximation to the Cristoffel terms assuming local‐boundary orthogonality, and another designed to procure boundary orthogonality. The performance of the new scheme is shown to be comparable with that of conventional inverse methods when calculations are performed on benchmark problems through the application of point‐by‐point and whole‐field solution schemes. Advantages and disadvantages of the present method are critically appraised. Copyright © 1999 National Research Council of Canada.     

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.     

On the composite Riemann problem for multi‐material fluid flows

We develop an Eulerian fixed grid numerical method for calculating multi‐material fluid flows. This approach relates to the class of interface capturing methods. The fluid is treated as a heterogeneous mixture of constituent materials, and the material interface is implicitly captured by a region of mixed cells that have arisen owing to numerical diffusion. To suppress this numerical diffusion, we propose a composite Riemann problem (CRP), which describes the decay of an initial discontinuity in the presence of a contact point between two different fluids, which is located off the initial discontinuity point. The solution to the CRP serves to calculate multi‐material no mixed numerical flux without introducing any material diffusion. We discuss the CRP solution and its implementation in the multi‐material fluid Godunov method. Numerical results show that a simple framework of the CRP greatly improves capturing material interfaces in the Godunov method and reproduces many of the advantages of more complicated interface tracking multi‐material treatments. Copyright © 2014 John Wiley & Sons, Ltd.     

Pressure‐stabilized maximum‐entropy methods for incompressible Stokes

We present a parameter‐free stable maximum‐entropy method for incompressible Stokes flow. Derived from a least‐biased optimization inspired by information theory, the meshfree maximum‐entropy method appears as an interesting alternative to classical approximation schemes like the finite element method. Especially compared with other meshfree methods, e.g. the moving least‐squares method, it allows for a straightforward imposition of boundary conditions. However, no Eulerian approach has yet been presented for real incompressible flow, encountering the convective and pressure instabilities. In this paper, we exclusively address the pressure instabilities caused by the mixed velocity‐pressure formulation of incompressible Stokes flow. In a preparatory discussion, existing stable and stabilized methods are investigated and evaluated. This is used to develop different approaches towards a stable maximum‐entropy formulation. We show results for two analytical tests, including a presentation of the convergence behavior. As a typical benchmark problem, results are also shown for the leaky lid‐driven cavity. The already presented information‐flux method for convection‐dominated problems in mind, we see this as the last step towards a maximum‐entropy method capable of simulating full incompressible flow problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Depth‐integrated,non‐hydrostatic model with grid nesting for tsunami generation,propagation, and run‐up

Tsunamis generated by earthquakes involve physical processes of different temporal and spatial scales that extend across the ocean to the shore. This paper presents a shock‐capturing dispersive wave model in the spherical coordinate system for basin‐wide evolution and coastal run‐up of tsunamis and discusses the implementation of a two‐way grid‐nesting scheme to describe the wave dynamics at resolution compatible to the physical processes. The depth‐integrated model describes dispersive waves through the non‐hydrostatic pressure and vertical velocity, which also account for tsunami generation from dynamic seafloor deformation. The semi‐implicit, finite difference model captures flow discontinuities associated with bores or hydraulic jumps through the momentum‐conserved advection scheme with an upwind flux approximation. The two‐way grid‐nesting scheme utilizes the Dirichlet condition of the non‐hydrostatic pressure and both the horizontal velocity and surface elevation at the inter‐grid boundary to ensure propagation of dispersive waves and discontinuities across computational grids of different resolution. The inter‐grid boundary can adapt to bathymetric features to model nearshore wave transformation processes at optimal resolution and computational efficiency. A coordinate transformation enables application of the model to small geographic regions or laboratory experiments with a Cartesian grid. A depth‐dependent Gaussian function smoothes localized bottom features in relation to the water depth while retaining the bathymetry important for modeling of tsunami transformation and run‐up. Numerical experiments of solitary wave propagation and N‐wave run‐up verify the implementation of the grid‐nesting scheme. The 2009 Samoa Tsunami provides a case study to confirm the validity and effectiveness of the modeling approach for tsunami research and impact assessment. Copyright © 2010 John Wiley & Sons, Ltd.     

Hybrid and multi‐point formulations of the lowest‐order mixed methods for Darcy's flow on triangles

Mixed finite element (MFE) and multipoint flux approximation (MPFA) methods have similar properties and are well suited for the resolution of Darcy's flow on anisotropic and heterogeneous domains. In this work, the link between hybrid and MPFA formulations is shown algebraically for the lowest order mixed methods of Raviart–Thomas (RT0) and Brezzi–Douglas–Marini (BDM1) on triangles. The efficiency of the four mixed formulations (Hybrid_RT0, MPFA_RT0, Hybrid_BDM1 and MPFA_BDM1) is investigated on high anisotropic and heterogeneous media and for unstructured triangular discretizations. Numerical experiments show that the MPFA_BDM1 formulation outperforms both Hybrid_RT0 and Hybrid_BDM1 in the case of anisotropic domains and highly unstructured meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

Application of the finite point method to high‐Reynolds number compressible flow problems

In this work, the finite point method is applied to the solution of high‐Reynolds compressible viscous flows. The aim is to explore this important field of applications focusing on two main aspects: the easiness and automation of the meshless discretization of viscous layers and the construction of a robust numerical approximation in the highly stretched clouds of points resulting in such domain areas. The flow solution scheme adopts an upwind‐biased scheme to solve the averaged Navier–Stokes equations in conjunction with an algebraic turbulence model. The numerical applications presented involve different attached boundary layer flows and are intended to show the performance of the numerical technique. The results obtained are satisfactory and indicative of the possibilities to extend the present meshless technique to more complex flow problems. Copyright © 2014 John Wiley & Sons, Ltd.     

A comparative study of direct‐forcing immersed boundary‐lattice Boltzmann methods for stationary complex boundaries

In this study, we assess several interface schemes for stationary complex boundary flows under the direct‐forcing immersed boundary‐lattice Boltzmann methods (IB‐LBM) based on a split‐forcing lattice Boltzmann equation (LBE). Our strategy is to couple various interface schemes, which were adopted in the previous direct‐forcing immersed boundary methods (IBM), with the split‐forcing LBE, which enables us to directly use the direct‐forcing concept in the lattice Boltzmann calculation algorithm with a second‐order accuracy without involving the Navier–Stokes equation. In this study, we investigate not only common diffuse interface schemes but also a sharp interface scheme. For the diffuse interface scheme, we consider explicit and implicit interface schemes. In the calculation of velocity interpolation and force distribution, we use the 2‐ and 4‐point discrete delta functions, which give the second‐order approximation. For the sharp interface scheme, we deal with the exterior sharp interface scheme, where we impose the force density on exterior (solid) nodes nearest to the boundary. All tested schemes show a second‐order overall accuracy when the simulation results of the Taylor–Green decaying vortex are compared with the analytical solutions. It is also confirmed that for stationary complex boundary flows, the sharper the interface scheme, the more accurate the results are. In the simulation of flows past a circular cylinder, the results from each interface scheme are comparable to those from other corresponding numerical schemes. Copyright © 2010 John Wiley & Sons, Ltd.