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

Zhang, Xiaoping; Su, Shuai; Wu, Jiming

doi:10.1016/j.jcp.2017.04.070

Cited by 41 publications

(26 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By (18) and (19) and the definitions of F n,1 K, and F n,1 L, , we have n K, F n,1 K, − n L, F n,1 L, = (1 − )…”

Section: The Finite Volume Schemementioning

confidence: 99%

“…We can see that the percentage of harmonic averaging points outside mesh edges is very low. In this case, we can still use the interpolation formula (19). In addition, the numerical scheme can also have expected accuracy as shown in Table 1 and Figure 3.…”

Section: Mild Anisotropymentioning

confidence: 99%

“…A close related concept is the so‐called monotone scheme that preserves the nonnegativity of the numerical solution, a special case of the minimum principle. In the context of anisotropic thermal conduction, the scheme without extremum preserving may lead the temperature to be negative in regions of large temperature variations . In order to avoid such non‐physical oscillations, extremum preserving is regarded as an indispensable requirement in constructing discrete schemes.…”

Section: Introductionmentioning

confidence: 99%

“…In the context of anisotropic thermal conduction, the scheme without extremum preserving may lead the temperature to be negative in regions of large temperature variations. 18,19 In order to avoid such non-physical oscillations, extremum preserving is regarded as an indispensable requirement in constructing discrete schemes.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A stabilized extremum‐preserving scheme for nonlinear parabolic equation on polygonal meshes

Peng

Gao

Feng

2019

Numerical Methods in Fluids

View full text Add to dashboard Cite

Summary In this paper, a stabilized extremum‐preserving scheme is introduced for the nonlinear parabolic equation on polygonal meshes. The so‐called harmonic averaging points located at the interface of heterogeneity are employed to define the auxiliary unknowns and can be interpolated by the cell‐centered unknowns. This scheme has only cell‐centered unknowns and possesses a small stencil. A stabilized term is constructed to improve the stability of this scheme. The stability analysis of this scheme is obtained under standard assumptions. Numerical results illustrate that the scheme satisfies the extremum principle with anisotropic full tensor coefficient problems and has optimal convergence rate in space on distorted meshes.

show abstract

“…By (18) and (19) and the definitions of F n,1 K, and F n,1 L, , we have n K, F n,1 K, − n L, F n,1 L, = (1 − )…”

Section: The Finite Volume Schemementioning

confidence: 99%

Section: Mild Anisotropymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A stabilized extremum‐preserving scheme for nonlinear parabolic equation on polygonal meshes

Peng

Gao

Feng

2019

Numerical Methods in Fluids

View full text Add to dashboard Cite

show abstract

“…Recently, by employing both cell-centered and vertex-centered unknowns as primary ones, Camier and Hermeline 10 suggested a new positivity-preserving scheme to avoid the issue of positivity-preserving interpolation; however, the convex decomposition of the co-normal vectors therein leads to the reduction of accuracy in the case of discontinuous diffusion coefficients. 11 A new nonlinear TPFA different from that of Potier 3 was studied in Zhang et al 11 and Gao and Wu,12 where both the convex decomposition of the co-normal vectors and the positivity-preserving interpolation of the auxiliary unknowns are not required, but the scheme must be designed very carefully to ensure a second-order truncation error. Here, we mention some related topics such as the extremum-preserving FV schemes 13,14 and the algebraic approaches.…”

mentioning

confidence: 99%

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

Dong

2018

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

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

show abstract

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

Peng

Gao

Feng

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 41 publications

References 30 publications

A stabilized extremum‐preserving scheme for nonlinear parabolic equation on polygonal meshes

A stabilized extremum‐preserving scheme for nonlinear parabolic equation on polygonal meshes

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

Contact Info

Product

Resources

About