Nonlinear Two-point Flux Approximation Research Articles

Error estimate of the cell-centered nonlinear positivity-preserving two-point flux approximation schemes

We present the error estimate of nonlinear positivity-preserving finite volume schemes for diffusion problems. These schemes are also named by nonlinear two-point flux approximation (NTPFA) methods. Due to their conservation and monotonicity, such schemes have attracted wide attention. Under suitable regularity assumptions on the exact solution, we prove a discrete H1 error estimate through the analysis of consistent errors. Some numerical examples are presented to demonstrate the theoretical results.

A positivity-preserving RD-FV scheme for diffusion problems on triangular meshes

In this paper, we propose a positivity-preserving residual distribution based finite volume scheme (RD-FV) for steady diffusion problems on unstructured triangular meshes. In this scheme, we use the RD method to obtain the auxiliary values of vertex, then adopt the positivity-preserving finite volume scheme to update the cell-centered values for diffusion problems, in which a nonlinear two-point flux approximation is utilized to obtain nonnegative solution. Thus a positivity-preserving conservative property for the RD-FV scheme is guaranteed. Several numerical experiments show the optimal convergence rates for the numerical solution, that is, approximate second-order accuracy for the solution and first-order accuracy for the flux, and demonstrate the positivity-preserving property of our new scheme.

The VEM-based positivity-preserving conservative scheme for radiation diffusion problems on generalized polyhedral meshes

In this paper, for the 3D radiation diffusion equation on generalized polyhedral meshes, we construct a positivity-preserving conservative scheme based on the virtual element method (VEM). Firstly, the equation is discretized by finite volume, and the one-sided flux on the cell-face is discretized by using the fixed stencil of all vertices. Then, the nonlinear two-point flux approximation is used to obtain the equations about cell-centered unknowns and vertex unknowns; the lowest order VEM solves the vertex values of the mesh. Finally, the conservative equations containing only cell-centered unknowns are solved. This scheme ensures the high accuracy of VEM and embodies the local conservation and positivity-preserving. Compared with the existing nonlinear positivity-preserving finite volume schemes, the new scheme does not need nonlinear iteration for linear problems. For nonlinear problems, it only needs to use the VEM to solve the nonlinear system. Numerical results show that the new scheme can achieve the optimal convergence order and maintain the nonnegativity of the solution for generalized polyhedral meshes and arbitrary diffusion tensors.

A Positivity-Preserving Finite Volume Scheme for Nonequilibrium Radiation Diffusion Equations on Distorted Meshes.

In this paper, we propose a new positivity-preserving finite volume scheme with fixed stencils for the nonequilibrium radiation diffusion equations on distorted meshes. This scheme is used to simulate the equations on meshes with both the cell-centered and cell-vertex unknowns. The cell-centered unknowns are the primary unknowns, and the element vertex unknowns are taken as the auxiliary unknowns, which can be calculated by interpolation algorithm. With the nonlinear two-point flux approximation, the interpolation algorithm is not required to be positivity-preserving. Besides, the scheme has a fixed stencil and is locally conservative. The Anderson acceleration is used for the Picard method to solve the nonlinear systems efficiently. Several numerical results are also given to illustrate the efficiency and strong positivity-preserving quality of the scheme.

A New Nonlinear Two-Point Flux Approximation Method for Solving the Anisotropic Diffusion Equation with Reduced Violations of the Discrete Maximum/Minimum Principle

Summary A class of monotone cell-centered nonlinear finite-volume methods has been proposed in the past decade to solve the anisotropic diffusion equation. The nonlinear two-point flux approximation (TPFA) (NTPFA) method preserves the nonnegativity of the solution values but can violate the discrete maximum/minimum principle (DMP). To enforce DMP, the nonlinear multipoint flux approximation (NMPFA) method ought to be used. In this work, we propose a novel NTPFA method that can reduce the severity of DMP violations significantly compared with the standard NTPFA method. The new formulation uses conormal decomposition for the construction of the one-sided fluxes. To define the unique flux through a connection between two cells, we choose a convex combination of the two one-sided fluxes such that the absolute differences of the magnitudes of the two transmissibility terms associated with the two neighboring cells are minimized, thus bringing the discrete coefficient matrix closer to having the zero row-sum property. Numerical experiments are conducted to test the performance of the new NTPFA method. The results demonstrate that the new scheme has comparable convergence order for both the solution and the flux compared with the standard NTPFA method or the classical multi-point flux approximation (MPFA-O) method. Moreover, the new NTPFA formulation shows marked improvements over the standard NTPFA in terms of reducing DMP violations. However, depending on the specific problem configuration, our new NTPFA formulation can lead to a system of nonlinear equations that is more difficult to solve.

A nonlinear correction scheme for the heterogeneous and anisotropic diffusion problems on polygonal meshes

In this paper, a nonlinear positivity-preserving finite volume scheme for the heterogeneous and anisotropic diffusion problems is proposed. Firstly, a linear diamond finite volume scheme is introduced. This scheme has both cell-centered unknowns and vertex unknowns. The vertex unknowns are treated as auxiliary ones and are expressed as linear combinations of the surrounding cell-centered unknowns, which reduces the scheme to a pure cell-centered one. Two new interpolation algorithms are suggested through the linearity-preserving approach and a novel discretization of diffusion coefficient. The new vertex interpolation algorithms improve a great deal the numerical performance of the linear scheme on distorted meshes with strongly anisotropy. Secondly, a new positivity-preserving cell-centered scheme is constructed via the aforementioned linear scheme and a nonlinear correction technique proposed by [28]. This nonlinear scheme is different from most existing nonlinear two-point flux approximation schemes in the way that it is not based on the nonlinear two-point flux approximation and it doesn't require the positivity-preserving vertex interpolation. Its implementation is also very easy and simple. Numerical experiments conducted in this paper indicate that the new positivity-preserving scheme is efficient and has approximately second order accuracy in most extreme cases.

Toward Accurate Reservoir Simulations on Unstructured Grids: Design of Simple Error Estimators and Critical Benchmarking of Consistent Discretization Methods for Practical Implementation

Summary In this paper, we aim to identify discretization errors caused by non-K-orthogonal grids upfront through simple preprocessing tools and perform a comparative study of a set of representative, state-of-the-art, consistent discretizations [multipoint flux approximation (MPFA-O), mimetic finite difference (MFD), nonlinear two-point flux approximation (NTPFA, TPFA), and average multipoint flux approximation (AvgMPFA)] to select the method most suited for inclusion in a commercial reservoir simulator. To predict the potential impact of discretization errors, we propose two types of error indicators. Static indicators measure the degree of nonconsistency of the two-point method at a cell level, and dynamic indicators measure how local discretization errors affect flow paths. The latter are computed using a series of idealized tracer simulations. By changing monitoring and injection points, one can mimic the reservoir-development strategy and thus focus on the errors introduced on quantities of real interest. To assess the practical usability of various consistent methods and validate our new error indicators, we use a set of representative grid models generated by contemporary commercial tools, for which we discuss static error indicators and compare tracer responses for the various discretization methods. We also compare degrees of freedom, sparsity, and the condition number of the alternative methods and discuss challenges related to their practical implementation. Our results indicate that tracer simulations constitute an efficient tool to identify and classify discretization errors and quantify their potential impact. We observe distinctively different behavior with the inconsistent two-point method and the consistent methods, which agree closely in terms of accuracy of the response. We also note a deficiency in the commercial realization of so-called Depogrids, which can result in unnecessarily complicated polytopal cells with hundreds of faces. Our overall conclusion is that NTPFA and AvgMPFA are the most viable solutions for integration into a commercial simulator, with the linear AvgMPFA method being the least invasive. NOTE: This paper is published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

The cell-centered positivity-preserving finite volume scheme for 3D anisotropic diffusion problems on distorted meshes

In this paper, a new cell-centered positivity-preserving finite volume scheme is proposed for the 3D anisotropic diffusion problems on distorted meshes. The primary unknowns and auxiliary unknowns are used in this scheme. First, the one-sided flux on each cell facet is discretized by a fixed stencil. Then, the nonlinear two-point flux approximation is applied to get the cell-centered discrete scheme. Besides, a second-order interpolation algorithm is constructed to eliminate the auxiliary unknowns in flux expressions. Furthermore, an improved Anderson acceleration method is given to accelerate the convergence of Picard iterations. Finally, several numerical examples are presented to show the accuracy and efficiency of the proposed scheme.

A positivity-preserving finite volume scheme for convection–diffusion equation on general meshes

A new positivity-preserving finite volume scheme is proposed for solving the convection–diffusion equation on 2D or 3D distorted meshes. The nonlinear two-point flux approximation is applied to the discretization of diffusion flux. The discretization of convection flux is based on the second-order upwind method with a slope limiter. The cell centres are employed to define the primary unknowns. The cell vertexes are devoted to define the auxiliary unknowns, which can be computed from the primary unknowns. And the interpolation method of this scheme is not required to be positivity-preserving. Numerical results illustrate that the scheme is effective in solving the convection–diffusion problem and has second-order convergence rate on the distorted meshes.

A non-linear finite volume method coupled with a modified higher order MUSCL-type method for the numerical simulation of two-phase flows in non-homogeneous and non-isotropic oil reservoirs

In this work, we propose a finite volume scheme to simulate two-phase flows in non-homogeneous and non-isotropic 2-D petroleum reservoirs. The governing equations are solved using the IMPES (IMplicit Pressure and Explicit Saturation) procedure, where the face fluxes from the pressure equation are approximated by a non-linear two-point flux approximation (NL-TPFA) that guarantees monotone solutions for the absolute pressure field. The scheme is based on the construction of one-sided fluxes on each adjacent cell independently and then, the unique edge flux is built as a linear combination of the one-sided fluxes. In our NL-TPFA finite volume scheme, we use auxiliary variables that are located on the vertices of the primal mesh. The nodal auxiliary unknowns are written as linear combinations of the neighboring cell-centered unknowns reducing our scheme to a fully cell-centered one. To solve the non-linear system of equations and to guarantee monotone solutions for arbitrarily anisotropic permeability tensors, we use the Picard iteration method with the Anderson acceleration technique to improve the computational efficiency. On the other hand, to solve the hyperbolic saturation equation, we propose a modified second-order finite volume method. The basic idea of our method is that the reconstructed saturation on the edge that violates the local Discrete Maximum Principle (DMP) is limited, otherwise, the reconstructed saturation is expressed as a convex combination of its unlimited and limited reconstructed values.

A Computational Workflow for Flow and Transport in Fractured Porous Media Based on a Hierarchical Nonlinear Discrete Fracture Modeling Approach

Modeling flow and transport in fractured porous media has been a topic of intensive research for a number of energy- and environment-related industries. The presence of multiscale fractures makes it an extremely challenging task to resolve accurately and efficiently the flow dynamics at both the local and global scales. To tackle this challenge, we developed a computational workflow that adopts a two-level hierarchical strategy based on fracture length partitioning. This was achieved by specifying a partition length to split the discrete fracture network (DFN) into small-scale fractures and large-scale fractures. Flow-based numerical upscaling was then employed to homogenize the small-scale fractures and the porous matrix into an equivalent/effective single medium, whereas the large-scale fractures were modeled explicitly. As the effective medium properties can be fully tensorial, the developed hierarchical framework constructed the discrete systems for the explicit fracture–matrix sub-domains using the nonlinear two-point flux approximation (NTPFA) scheme. This led to a significant reduction of grid orientation effects, thus developing a robust, applicable, and field-relevant framework. To assess the efficacy of the proposed hierarchical workflow, several numerical simulations were carried out to systematically analyze the effects of the homogenized explicit cutoff length scale, as well as the fracture length and orientation distributions. The effect of different boundary conditions, namely, the constant pressure drop boundary condition and the linear pressure boundary condition, for the numerical upscaling on the accuracy of the workflow was investigated. The results show that when the partition length is much larger than the characteristic length of the grid block, and when the DFN has a predominant orientation that is often the case in practical simulations, the workflow employing linear pressure boundary conditions for numerical upscaling give closer results to the full-model reference solutions. Our findings shed new light on the development of meaningful computational frameworks for highly fractured, heterogeneous geological media where fractures are present at multiple scales.

Convergence of nonlinear finite volume schemes for two-phase porous media flow on general meshes

Abstract In this work we present an abstract finite volume discretization framework for incompressible immiscible two-phase flow through porous media. A priori error estimates are derived that allow us to prove the existence of discrete solutions and to establish the proof of convergence for schemes belonging to this framework. In contrast to existing publications the proof is not restricted to a specific scheme and it assumes neither symmetry nor linearity of the flux approximations. Two nonlinear schemes, namely a nonlinear two-point flux approximation and a nonlinear multipoint flux approximation, are presented, and some properties of these schemes, e.g. saturation bounds, are proven. Furthermore, the numerical behavior of these schemes (e.g. accuracy, coercivity, efficiency or saturation bounds) is investigated for different test cases for which the coercivity is checked numerically.

A novel cell-centered finite volume scheme with positivity-preserving property for the anisotropic diffusion problems on general polyhedral meshes

In this paper, we propose a novel cell-centered positivity-preserving finite volume scheme for the anisotropic diffusion problems. The discretization of diffusion flux is based on the standard nonlinear two-point flux approximation. The cell vertexes are employed to define auxiliary unknowns. A positivity-preserving vertex interpolation algorithm is constructed to get the value of auxiliary unknowns. This interpolation algorithm has almost second order convergence rate on the distorted meshes. Numerical results illustrate that our scheme is efficient and practical in solving the 3D anisotropic diffusion problems.

A positivity-preserving finite volume scheme for three-temperature radiation diffusion equations

In this paper, a positivity-preserving finite volume scheme is proposed for the two-dimensional three-temperature (2-D 3-T) radiation diffusion equations. The cell centers are given to define the primary unknowns. The cell vertexes are used to define auxiliary unknowns, which can be computed by the primary unknowns. The nonlinear two-point flux approximation is applied to the discretization of diffusion flux. This scheme is not necessary the interpolation method with positivity-preserving property. Besides, this scheme is local conservative and second-order accurate on the distorted meshes. The existence of discrete solution for this scheme is proved. The stability analysis is also obtained with some assumptions. Numerical experiments indicate that the finite volume scheme is robust and well positive in solving the 2-D 3-T radiation diffusion equations.

A positivity-preserving nonlinear finite volume scheme for radionuclide transport calculations in geological radioactive waste repository

Purpose This paper aims to consider numerical simulation for radionuclide transport calculations in geological radioactive waste repository. Design/methodology/approach The nonlinear two-point flux approximation is used to discretize the diffusion flux and has a fixed stencil. The cell-vertex unknowns are applied to define the auxiliary unknowns and can be interpolated by the cell-centered unknowns. The approximation of convection flux is based on the second-order upwind method with a slope limiter. Findings Numerical results illustrate that the positivity-preserving is satisfied in solving this convection-diffusion system and has a second-order convergence rate on the distorted meshes. Originality/value A new positivity-preserving nonlinear finite volume scheme is proposed to simulate the far-field model used in the geological radioactive waste repository. Numerical results illustrate that the positivity-preserving is satisfied in solving this convection-diffusion system and has a second-order convergence rate on the distorted meshes.

A non-negative and high-resolution finite volume method for the depth-integrated solute transport equation using an unstructured triangular mesh

This paper proposes a new high-resolution finite volume method for solving the two-dimensional (2D) solute transport equation using an unstructured mesh. A new simple r-factor algorithm is introduced into the Total Variation Diminishing flux limiter to achieve a more efficient yet accurate high-resolution scheme for solving the advection term. To avoid the physically-meaningless negative solutions resulted from using the Green–Gauss theorem, a nonlinear two-point flux approximation scheme is adopted to deal with the anisotropic diffusion term. The developed method can be readily coupled with a two-dimensional finite-volume-based flow models under unstructured triangular mesh. By integrating with the ELCIRC flow model, the proposed method was verified using three idealized benchmark cases (i.e., advection of a circle-shaped solute field, advection in a cyclogenesis flow field and transport of a initially square-shaped solute plume), and further applied to simulate the non-reactive solute transport process driven by irregular tides in the Deep Bay, eastern Pearl River Estuary of China. These cases are also simulated by models using other existing methods, including different r-factor for advection term and the Green–Gauss theorem for diffusion term. The comparison between the results from the new method and those from other existing methods demonstrated the new method could describe advection induced concentration shock and discontinuities, and anisotropic diffusion at high resolution without providing spurious oscillations and negative values.

Monotone nonlinear finite-volume method for challenging grids

This article presents a new positivity-preserving finite-volume scheme with a nonlinear two-point flux approximation, which uses optimization techniques for the face stencil calculation. The gradient is reconstructed using harmonic averaging points with the constraint that the sum of the coefficients included in the face stencils must be positive. We compare the proposed scheme to a nonlinear two-point scheme available in literature and a few linear schemes. Using two test cases, taken from the FVCA6 benchmarks, the accuracy of the scheme is investigated. Furthermore, it is shown that the scheme is linearity-preserving on highly complex corner-point grids. Moreover, a two-phase flow problem on the Norne formation, a geological formation in the Norwegian Sea, is simulated. It is demonstrated that the proposed scheme is consistent in contrast to the linear Two-Point Flux Approximation scheme, which is industry standard for simulating subsurface flow on corner-point grids.

A vertex-centered and positivity-preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids

We suggest a new positivity-preserving finite volume scheme for anisotropic diffusion problems on arbitrary polygonal grids. The scheme has vertex-centered, edge-midpoint and cell-centered unknowns. The vertex-centered unknowns are primary and have finite volume equations associated with them. The edge-midpoint and cell-centered unknowns are treated as auxiliary ones and are interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. Unlike most existing positivity-preserving schemes, the construction of the scheme is based on a special nonlinear two-point flux approximation that has a fixed stencil and does not require the convex decomposition of the co-normal. In order to solve efficiently the nonlinear systems resulting from the nonlinear scheme, Picard method and its Anderson acceleration are discussed. Numerical experiments demonstrate the second-order accuracy and well positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids. The high efficiency of the Anderson acceleration is also shown on reduction of the number of nonlinear iterations. Moreover, the proposed scheme does not have the so-called numerical heat-barrier issue suffered by most existing cell-centered and hybrid schemes. However, further improvements have to be made if the solution is very close to the machine precision and the mesh distortion is very severe.

A Second-Order Positivity-Preserving Finite Volume Scheme for Diffusion Equations on General Meshes

We propose a new positivity-preserving finite volume scheme for the anisotropic diffusion problems on general polygonal meshes based on a new nonlinear two-point flux approximation. The scheme uses both cell-centered and cell-vertex unknowns. The cell-vertex unknowns are treated as auxiliary ones and are computed by two second-order interpolation algorithms. Due to the new nonlinear two-point flux formulation, it is not required to replace the interpolation algorithm with positivity-preserving but usually low-order accurate ones whenever negative interpolation weights occur and it is also unnecessary to require the decomposition of the conormal vector to be a convex one. Moreover, the new nonlinear two-point flux approximation has a fixed stencil. These features make our scheme more flexible, easy for implementation, and different from other existing nonlinear schemes based on Le Potier's two-point flux approximation. The positivity-preserving property of our scheme is proved theoretically and numerical r...

Interpolation-based second-order monotone finite volume schemes for anisotropic diffusion equations on general grids

We propose two interpolation-based monotone schemes for the anisotropic diffusion problems on unstructured polygonal meshes through the linearity-preserving approach. The new schemes are characterized by their nonlinear two-point flux approximation, which is different from the existing ones and has no constraint on the associated interpolation algorithm for auxiliary unknowns. Thanks to the new nonlinear two-point flux formulation, it is no longer required that the interpolation algorithm should be a positivity-preserving one. The first scheme employs vertex unknowns as the auxiliary ones, and a second-order but not positivity-preserving interpolation algorithm is utilized. The second scheme uses the so-called harmonic averaging points located on cell edges to define the auxiliary unknowns, and a second-order positivity-preserving interpolation method is employed. Both schemes have nearly the same convergence rates as compared with their related second-order linear schemes. Numerical results demonstrate that the new schemes are monotone, and have the second-order accuracy for the solution and first-order for its gradient on severely distorted meshes.