Convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes

Schneider, Martin; Agélas, Léo; Enchéry, Guillaume; Flemisch, Bernd

doi:10.1016/j.jcp.2017.09.003

Cited by 35 publications

(42 citation statements)

References 32 publications

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“…Therefore, appropriate schemes have to be chosen application dependent. In general, the convergence of schemes can be proven if the scheme is consistent and coercive, as it has been done in [4,8,12,21]. In the following, we briefly describe such fundamental properties for cell-centered schemes (with discrete solution space (12)) and for HMM schemes (with discrete solution space (14)).…”

Section: Properties Of Discretization Schemesmentioning

confidence: 99%

“…where D & C 0 ðXÞ is a test function space which is assumed to be dense in H 1 0 ðXÞ, and F K ;r ; F K;r are the discrete and exact flux functions, respectively. Furthermore, u D ¼ ðu T ; u E Þ 2 X D is defined as ðu T Þ K ¼ uðx K Þ for all K 2 T , and ðu E Þ r ¼ uðx r Þ for all r 2 E. More details can for example be found in [21].…”

Section: Consistencymentioning

confidence: 99%

“…However, this comes with the cost of additional unknowns. Recently, monotone or discrete extremum-principles-preserving schemes have been developed in [17][18][19][20][21][22], and applied to highly complex porous media applications in [23][24][25].…”

Section: Introductionmentioning

confidence: 99%

“…For a more detailed definition of an admissible discretization see [3,4,21]. Let M be any set, then the cardinality of this set is indicated with n M .…”

Section: Introductionmentioning

confidence: 99%

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Comparison of finite-volume schemes for diffusion problems

Schneider

Gläser

Flemisch

et al. 2018

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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We present an abstract discretization framework and demonstrate that various cell-centered and hybrid finite-volume schemes fit into it. The different schemes considered in this work are then analyzed numerically for an elliptic model problem with respect to the properties consistency, coercivity, extremum principles, and sparsity. The test cases presented comprise of two-and three-dimensional setups, mildly and highly anisotropic tensors and grids of different complexities. The results show that all schemes show a similar convergence behavior, except for the two-point flux approximation scheme, and seem to be coercive. Furthermore, they confirm that linear schemes, in contrast to nonlinear schemes, are in general neither positivity-preserving nor satisfy discrete minimum or maximum principles. Discretization schemesLet X & R d ; d 2 N Ã , be an open bounded connected polygonal domain with boundary oX, and d-dimensional measure |X|. In the following, we consider the elliptic problem r Á ÀKru ð Þ¼f in X;Here, we assume that f 2 L 2 (X) and K is a symmetric tensor-valued function such that (s.t.) the spectrum of K(x) is contained in [a 0 , b 0 ], with 0 < a 0 < b 0 < +1, for

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Section: Properties Of Discretization Schemesmentioning

confidence: 99%

Section: Consistencymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For a more detailed definition of an admissible discretization see [3,4,21]. Let M be any set, then the cardinality of this set is indicated with n M .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Comparison of finite-volume schemes for diffusion problems

Schneider

Gläser

Flemisch

et al. 2018

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

Self Cite

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“…The accuracy in the mildly anisotropic problem on 3D Kershaw meshes, ie, mesh D in Figure 6, is not so good as we expected and should be improved later. Up to now, the existence or convergence of the numerical solution for some nonlinear cell-centered FV schemes (see, eg, Blanc and Labourasse 6 and Schneider et al 46 ) has been analyzed. In the future, we will focus on these aspects for our vertex-centered positivity-preserving scheme.…”

Section: Discussionmentioning

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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An extremum‐preserving finite volume scheme for three‐temperature radiation diffusion equations

Peng

Gao

Feng

2022

Math Methods in App Sciences

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In this paper, an extremum‐preserving finite volume scheme is constructed for the two‐dimensional three‐temperature (2D 3‐T) radiation diffusion equations. The harmonic averaging points located at cell edge are applied to define the auxiliary unknowns, and the primary unknowns are defined at cell center. This scheme has a fixed stencil and satisfies the local conservation condition and discrete extremum principle. The existence of discrete solution is proved by using the fixed point theorem. Moreover, the stability analysis of this scheme is also presented. Numerical results illustrate that this scheme is efficient and accurate in solving the 2D 3‐T radiation diffusion equations on distorted meshes.

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Convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes

Cited by 35 publications

References 32 publications

Comparison of finite-volume schemes for diffusion problems

Comparison of finite-volume schemes for diffusion problems

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

An extremum‐preserving finite volume scheme for three‐temperature radiation diffusion equations

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