A monotone nonlinear finite volume method for diffusion equations and multiphase flows

Nikitin, Kirill D.; Terekhov, Kirill M.; Vassilevski, Yuri

doi:10.1007/s10596-013-9387-6

Cited by 70 publications

(13 citation statements)

References 20 publications

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“…The last characters in the names of the different MPFA methods are related to the shapes of the stencils used for flux computation. For dealing with a full permeability tensor while maintaining the simplicity of the two-point flux approximation method, a nonlinear two-point flux approximation method was developed (Chen et al, 2008, Nikitin et al, 2014 using a nonlinear relation for calculating flux between the two points. The mimetic finite difference method (Lipnikov et al, 2014) can also model flow through porous or fractured media with a full permeability tensor.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

Modeling anisotropic flow and heat transport by using mimetic finite differences

Chen

Clauser

Marquart

et al. 2016

Advances in Water Resources

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Section: Accepted Manuscriptmentioning

confidence: 99%

Modeling anisotropic flow and heat transport by using mimetic finite differences

Chen

Clauser

Marquart

et al. 2016

Advances in Water Resources

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“…Most of the existing literature about nonlinear finite‐volume schemes focuses on linear elliptic equations. Only a few publications exist that consider multi‐phase flow in porous media . However, they do not account for compositional and nonisothermal effects.…”

Section: Introductionmentioning

confidence: 99%

“…Most of the existing literature about nonlinear finite-volume schemes focuses on linear elliptic equations. Only a few publications exist that consider multi-phase flow in porous media [31,32]. However, they do not account for compositional and nonisothermal effects.A main constituent of this work is the numerical analysis of accuracy and efficiency of a nonlinear finite-volume scheme for the nonisothermal two-phase two-component flow equations.…”

mentioning

confidence: 99%

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

Schneider

Flemisch

Helmig

2016

Numerical Methods in Fluids

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Summary 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.

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“…Following the idea of Potier, 3 Kapyrin 19 proposed a family of positivity-preserving schemes on unstructured tetrahedral meshes. Under the framework of Lipnikov et al, 9 positivity-preserving FV schemes for diffusion equations were developed in Danilov and Vassilevski 20,21 on conformal polyhedral meshes with planar faces, and this method was further extended to advection-diffusion equations 22 and multiphase flow model 23 on unstructured polyhedral meshes. In addition, piecewise linear transformation was proposed in Vidović et al 18 to obtain a complicated 3D positivity-preserving interpolation method, and it was extended in another study 24 to discretize the diffusive flux under the assumption that the structure of the domain is locally layered.…”

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confidence: 99%

A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes

Dong

2018

Math Methods in App Sciences

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We propose a new nonlinear positivity-preserving finite volume scheme for anisotropic diffusion problems on general polyhedral meshes with possibly nonplanar faces. The scheme is a vertex-centered one where the edgecentered, face-centered, and cell-centered unknowns are treated as auxiliary ones that can be computed by simple second-order and positivity-preserving interpolation algorithms. Different from most existing positivity-preserving schemes, the presented 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. More interesting is that the flux discretization is actually performed on a fixed tetrahedral subcell of the primary cell, which makes the scheme very easy to be implemented on polyhedral meshes with star-shaped cells. Moreover, it is suitable for polyhedral meshes with nonplanar faces, and it does not suffer the so-called numerical heat-barrier issue. The truncation error is analyzed rigorously, while the Picard method and its Anderson acceleration are used for the solution of the resulting nonlinear system.Numerical experiments are also provided to demonstrate the second-order accuracy and well positivity of the numerical solution for heterogeneous and anisotropic diffusion problems on severely distorted grids. KEYWORDS diffusion equation, nonlinear two-point flux approximation, positivity-preserving, vertex-centered scheme 1 | INTRODUCTIONDiffusion equations are required to be solved efficiently and robustly on arbitrary distorted polygonal or polyhedral meshes in the numerical modelling of many physical problems, such as fluid flows in complex reservoir models, Lagrangian hydrodynamics with heat and radiative diffusion, and Lagrangian magnetohydrodynamics with magnetic diffusion. In some applications such as radiation hydrodynamics, the discrete solutions of diffusion equations should be nonnegative in order to avoid nonphysical quantities. The positivity-preserving or monotone property is one of the key requirements for diffusion schemes. In recent years, the study of positivity-preserving finite volume (FV) discretizations for diffusion problems has drawn great attention and many efforts have been devoted to this area.Since linear diffusion FV schemes do not unconditionally satisfy the positivity-preserving property and simultaneously provide a good approximation on distorted meshes or with high anisotropy ratio, 1,2 many researchers turn to

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A monotone nonlinear finite volume method for diffusion equations and multiphase flows

Cited by 70 publications

References 20 publications

Modeling anisotropic flow and heat transport by using mimetic finite differences

Modeling anisotropic flow and heat transport by using mimetic finite differences

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

A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes

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