Control Volume Finite Element Scheme Research Articles

Enhanced positive vertex-centered finite volume scheme for anisotropic convection-diffusion equations

This article is about the development and the analysis of an enhanced positive control volume finite element scheme for degenerate convection-diffusion type problems. The proposed scheme involves only vertex unknowns and features anisotropic fields. The novelty of the approach is to devise a reliable upwind approximation with respect to flux-like functions for the elliptic term. Then, it is shown that the discrete solution remains nonnegative. Under general assumptions on the data and the mesh, the convergence of the numerical scheme is established owing to a recent compactness argument. The efficiency and stability of the methodology are numerically illustrated for different anisotropic ratios and nonlinearities.

Convergence of a positive nonlinear control volume finite element scheme for an anisotropic seawater intrusion model with sharp interfaces

AbstractWe study a sharp interface model in the context of seawater intrusion in an anisotropic unconfined aquifer. It is a degenerate parabolic system with cross‐diffusion modeling the flow of fresh and saltwater. We study a nonlinear control volume finite element scheme. This scheme ensures the nonnegativity of the discrete solution without any restriction on the transmissibility coefficients. Moreover, it also provides a control on the entropy. The existence of a discrete solution and the convergence of this scheme are obtained, based on nonlinear stability results.

Positive control volume finite element scheme for a degenerate compressible two-phase flow in anisotropic porous media

In this paper, we are concerned with the convergence analysis of a positive control volume finite element scheme (CVFE) for a degenerate compressible two-phase flow model in anisotropic porous media. For this, we consider the global pressure saturation formulation. We next use an implicit Euler scheme in time and a CVFE discretization in space. This approach rests on a particular choice of the mean value of the gas density on the interfaces, a centered scheme of the total mobility, and the upwind approximation of fractional fluxes according to the total velocity. Thus, the maximum principle is fulfilled without any constraint on the stiffness coefficients. Moreover, uniform estimates on the discrete gradient of the global pressure and the dissipative term are derived. As the mesh size is sent to zero, we establish that the sequence of approximate solutions converges to a weak solution of the continuous problem. Numerical tests are presented in two dimensions to exhibit the behavior of the gas pressure and the water saturation through the medium.

Numerical analysis of a nonlinearly stable and positive control volume finite element scheme for Richards equation with anisotropy

We extend the nonlinear Control Volume Finite Element scheme of [C. Cancès and C. Guichard, Math. Comput. 85 (2016) 549–580]. to the discretization of Richards equation. This scheme ensures the preservation of the physical bounds without any restriction on the mesh and on the anisotropy tensor. Moreover, it does not require the introduction of the so-called Kirchhoff transform in its definition. It also provides a control on the capillary energy. Based on this nonlinear stability property, we show that the scheme converges towards the unique solution to Richards equation when the discretization parameters tend to 0. Finally we present some numerical experiments to illustrate the behavior of the method.

Convergence of a positive nonlinear Control Volume Finite Element scheme for solving an anisotropic degenerate breast cancer development model

In this paper, a nonlinear control volume finite element (CVFE) scheme for solving an anisotropic degenerate breast cancer development model is introduced and analyzed. This model includes both ordinary differential equations and convection–diffusion–reaction equations modeling the stepwise mutations from a normal breast stem cell to a tumor cell. The diffusion term, which generally involves an anisotropic and heterogeneous diffusion tensor, is discretized on a dual mesh by means of the piecewise linear conforming finite element method and using the Godunov scheme to approximate the diffusion fluxes provided by the conforming finite element reconstruction. The other terms are discretized using a nonclassical upwind finite volume scheme on the dual mesh, where the dual volumes are constructed around the vertices of the original mesh. This technique ensures the positivity and boundedness of discrete solutions without any restriction on the diffusion tensor nor the transmissibility coefficients. The convergence of the scheme is proved, only supposing the shape regularity condition for the original mesh and using a priori estimates as well as the Kolmogorov relative compactness theorem. The proposed scheme is robust, locally conservative, efficient, and stable, which is confirmed by numerical experiments over a general mesh.

Immersed Boundary Method Based on the Implementation of Conservation Equations Along the Boundary Using Control-Volume Finite-Element Scheme

In this study conservation equations were implemented along the boundaries via ghost control-volume immersed boundary method. The control-volume finiteelement method was applied on a cartesian grid to simulate 2-D incompressible flow. In this approach, mass and momentum equations were conserved in the whole domain including boundary control volumes by introducing ghostcontrol volume concept. The Taylor problem was selected to validate the present method. Four different case studies of Taylor problem encompassing both inviscid and viscous flow conditions in ordinary and 45° rotated grid were used for more investigation. Comparisons were made between the results of the present method and those obtained from the exact solution. Results of the present method indicated accurate predictions of the velocity and pressure fields in midline, diagonal, and all boundaries. The agreement between the results of the present method and the exact solution was very good throughout the whole temporal domain. Furthermore, comparison of the rate of kinetic energy decay in viscous case showed same level of agreement between the results.

On the efficacy of a control volume finite element method for the capture of patterns for a volume-filling chemotaxis model

In this paper, a control volume finite element scheme for the capture of spatial patterns for a volume-filling chemotaxis model is proposed and analyzed. The diffusion term, which generally involves an anisotropic and heterogeneous diffusion tensor, is discretized by piecewise linear conforming triangular finite elements (P1-FEM). The other terms are discretized by means of an upstream finite volume scheme on a dual mesh, where the dual volumes are constructed around the vertices of each element of the original mesh. The scheme ensures the validity of the discrete maximum principle under the assumption that the transmissibility coefficients are nonnegative. The convergence analysis is based on the establishment of a priori estimates on the cell density, these estimates lead to some compactness arguments in L2 based on the use of the Kolmogorov compactness theorem. Finally, we show some numerical results to illustrate the effectiveness of the scheme to capture the pattern formation for the mathematical model.

Stability analysis and applications of the control volume finite element scheme for two phase flow in the petroleum reservoirs

In this paper, in order to design a useful computational flow simulation for the governing equation of two phase incompressible (water and oil), a robust and efficient scheme is proposed. This hybrid method consisting of IMPES technique for the linearisation, control volume finite element method on hexagonal grids for spatial discretisation, backward Euler method for time discretisation and stable variant (adaptive-simpler-GMRES) of simpler generalised minimum residual technique with incomplete factorisation preconditioners is used to solve the corresponding linear system. Moreover, we proposed the general Courant Friedrich Levy condition for two phase immiscible flow with Von Neumann method. Numerical results for oil pressure, oil and water production rates and water saturation in the injection and production wells are illustrated.

Krylov subspace approximations for the exponential Euler method: error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product $\varphi(tA)b$ where $\varphi(z) = [\exp(z)-1]/z$. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where $A$~is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to $\varphi(tA)b$, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error. References E. J. Carr, T. J. Moroney and I. W. Turner. Efficient simulation of unsaturated flow using exponential time integration. Appl. Math. Comput., 217(14):6587--6596, 2011. doi:10.1016/j.amc.2011.01.041 E. J. Carr, I. W. Turner and P. Perre. A Jacobian-free exponential integrator for simulating transport in heterogeneous porous media: application to the drying of softwood, submitted for publication. E. J. Carr, I. W. Turner and P. Perre. A new control-volume finite-element scheme for heterogeneous porous media:application to the drying of softwood. Chem. Eng. Technol., 34(7):1143--1150, 2011. doi:10.1002/ceat.201100060 E. Celledoni and I. Moret. A Krylov projection method for systems of ODEs. Appl. Numer. Math., 24(2--3):365--378, 1997. doi:10.1016/S0168-9274(97)00033-0 N. J. Higham. Functions of matrices: theory and computation. SIAM, Philadelphia, PA, USA, 2008. M. Hochbruck, M. E. Hochstenbach. Subspace extraction for matrix functions, submitted for publication. http://www.win.tue.nl/ hochsten/pdf/funext.pdf M. Hochbruck, C. Lubich and H. Selhofer. Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput., 19(5):1552--1574, 1998. doi:10.1137/S1064827595295337 B. V. Minchev and W. M. Wright. A review of exponential integrators for first order semi-linear problems. Numerics No. 2/05, Norwegian University of Science and Technology, Trondheim, Norway, 2005. http://www.math.ntnu.no/preprint/numerics/2005/N2-2005.ps P. Perre and I. Turner. A heterogeneous wood drying computational model that accounts for material property variation across growth rings. Chem. Eng. J., 86(1--2):117--131, 2002. doi:10.1016/S1385-8947(01)00270-4

SIMPLIFIED CONTROL-VOLUME FINITE-ELEMENT METHOD

Localized vector algebra treatment of nonorthogonality is applied to two-dimensional quadrilateral control volumes using Cartesian base vectors in a primitive variable formulation of the Navier-Stokes equations for steady incompressible laminar flow. With optional grid-aligned, locally analytic interpolation, a simplified control-volume finite-element scheme is presented. Discretization of source terms, determination of interface convection-diffusion fluxes, pressure correction factors, and geometric quantities are described briefly. Results of three test cases provide useful initial insights into the performance of the method. The conclusion is reached that a simple finite-volume-based approach to nonorthogonality has been achieved.

Mathematical and Numerical Properties of Control-Volume, Finite-Element Scheme for Reservoir Simulation

Summary This paper presents the mathematical properties of a control-volume, finite-element (CVFE) scheme. We show that appropriate constraints on finite-element grids and convenient definitions for volumes and transmissibilities lead to a convergence property for the CVFE scheme. This convergence property proves that use of the CVFE scheme is mathematically correct for reservoir simulation. With the control-volume concept, the local balances of each component are fully satisfied. Because the discretized equations resulting from the scheme can be solved by classic methods, the CVFE scheme can be implemented easily in a general-purpose simulator. In the implementations in this paper, we use prismatic finite elements for 3D full-field simulations and triangles for 2D simulations. The grid is generated with automatic techniques, popular in structural engineering, resulting in numerical diffusion that is as isotropic as possible; therefore, grid-orientation effects are controlled and accuracy is easily improved through natural grid refinement. Examples show that, for the same accuracy, computational costs are lower for results obtained with the CVFE scheme than for those obtained with finite-difference grids. The CVFE scheme is an excellent alternative to flexible gridding techniques used in finite-difference simulators because the entire reservoir can be gridded as required, without use of special techniques for local grid modifications.

A streamline upwind control volume finite element method for modeling fluid flow and heat transfer problems

Galerkin and control volume finite element approximations are compared and contrasted. Streamline upwinding and pressure stabilizing Petrov-Galerkin formulations for the numerical modeling of fluid flow problems are discussed. These formulations are then adapted to a control volume finite element discretization. The resulting control volume finite element scheme permits an equal order interpolation for velocity and pressure, is readily implemented and retains the property of local conservation, normally associated with control volume schemes.

STREAMLINE UPWIND SCHEME FOR CONTROL-VOLUME FINITE ELEMENTS, PART II. IMPLEMENTATION AND COMPARISON WITH THE SUPG FINITE-ELEMENT SCHEME

The streamline upwind control-volume (SUCV) finite-element scheme is compared to the streamline upwind Petrov Galerkin (SUPG) finite-element scheme for a range of steady-state and transient test problems. The results indicate that for steady-state problems, the two schemes perform at the same level; while for transient convective problems, the SUPG scheme is slightly better. In addition, a qualitative performance comparison is made with previous control-volume finite-element schemes. These comparisons indicate that the performance of the SUCV scheme is competitive.