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

Schneider, Martin; Flemisch, Bernd; Helmig, Rainer

doi:10.1002/fld.4352

Cited by 49 publications

(16 citation statements)

References 55 publications

(108 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An established idea to obtain monotone or extremum-principles-preserving schemes, as those developed in [15,16,17,18,20,21,24,19], is to compute for each interior edge σ ∈ E int , with…”

Section: Application To Some Nonlinear Finite Volume Schemesmentioning

confidence: 99%

See 1 more Smart Citation

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

Schneider

Agélas

Enchéry

et al. 2017

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

In the present work, we deal with the convergence of cell-centered nonlinear finite volume schemes for anisotropic and heterogeneous diffusion operators. A general framework for the convergence study of finite volume methods is provided and used to establish the convergence of the new methods. Thorough assessment on a set of anisotropic heterogeneous problems as well as a comparison with linear finite volume schemes is provided.

show abstract

“…An established idea to obtain monotone or extremum-principles-preserving schemes, as those developed in [15,16,17,18,20,21,24,19], is to compute for each interior edge σ ∈ E int , with…”

Section: Application To Some Nonlinear Finite Volume Schemesmentioning

confidence: 99%

“…The proof relies on concepts that have been developed in [4]. It generalizes the one given in [19] and allows to prove the convergence for the nonlinear finite volume schemes introduced in [15,16,17,18,20,21] for which no proof yet existed, as mentioned in [22].…”

Section: Introductionmentioning

confidence: 99%

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

Schneider

Agélas

Enchéry

et al. 2017

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…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%

Comparison of finite-volume schemes for diffusion problems

Schneider

Gläser

Flemisch

et al. 2018

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

Self Cite

View full text Add to dashboard Cite

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

show abstract

“…The kernel of a positivity-preserving FV scheme is the decomposition of the co-normal and the construction of the flux approximation. To our knowledge, most existing positivity-preserving FV schemes use the convex decomposition of the co-normal, which leads to a dynamic stencil for the flux approximation and requires a certain search algorithm (see [20,[35][36][37] and the references therein). Here we adopt a different approach to design a flux approximation with a fixed stencil, avoiding the search algorithm for complex grids.…”

Section: Introductionmentioning

confidence: 99%

A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

sci¹

2020

NMTMA

View full text Add to dashboard Cite

Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multimaterial system and explain the exchange of energy among electrons, ions and photons. In this paper, we suggest a new positivity-preserving finite volume scheme for 2-D 3-T radiation diffusion equations on general polygonal meshes. The vertex unknowns are treated as primary ones for which the finite volume equations are constructed. The edge-midpoint and cell-centered unknowns are used as auxiliary ones and interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. By comparison, most existing positivity-preserving finite volume schemes are cell-centered and based on the convex decomposition of the co-normal. Here, the co-normal decomposition is not convex in general, leading to a fixed stencil of the flux approximation and avoiding a certain search algorithm on complex grids. Moreover, the new scheme effectively alleviates the numerical heat-barrier issue suffered by most existing cell-centered or hybrid schemes in solving strongly nonlinear radiation diffusion equations. Numerical experiments demonstrate the second-order accuracy and the positivity of the solution on various distorted grids. For the problem without analytic solution, the contours of the numerical solutions obtained by our scheme on distorted meshes accord with those on smooth quadrilateral meshes.

show abstract

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

Cited by 49 publications

References 55 publications

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

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

Comparison of finite-volume schemes for diffusion problems

A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

Contact Info

Product

Resources

About