A decoupled and positivity-preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes

Su, Shuai; Dong, Qiannan; Wu, Jiming

doi:10.1016/j.jcp.2018.06.052

Cited by 28 publications

(12 citation statements)

References 50 publications

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“…We find that on Mesh1, Mesh3, Mesh4 and Mesh5, the optimal convergence rates for the solution and its flux are obtained, but the least square order of the solution errors on Mesh2 and Mesh3 are h 1.59 and h 1.35 respectively. The scheme LPEW2 performs similar results which can be (Su et al, 2018). Here we adopt a simple iterative Piccard scheme to solve the nonlinearity.…”

Section: Test 5: Nonlinear Problemmentioning

confidence: 89%

A linearity-preserving vertex interpolation algorithm for cell-centered finite volume approximations of anisotropic diffusion problems

Yang

Gao

2019

HFF

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Purpose A finite volume scheme for diffusion equations on non-rectangular meshes is proposed in [Deyuan Li, Hongshou Shui, Minjun Tang, J. Numer. Meth. Comput. Appl., 1(4)(1980)217–224 (in Chinese)], which is the so-called nine point scheme on structured quadrilateral meshes. The scheme has both cell-centered unknowns and vertex unknowns which are usually expressed as a linear weighted interpolation of the cell-centered unknowns. The critical factor to obtain the optimal accuracy for the scheme is the reconstruction of vertex unknowns. However, when the mesh deformation is severe or the diffusion tensor is discontinuous, the accuracy of the scheme is not satisfactory, and the author hope to improve this scheme. Design/methodology/approach The authors propose an explicit weighted vertex interpolation algorithm which allows arbitrary diffusion tensors and does not depend on the location of discontinuity. Both the derivation of the scheme and that of vertex reconstruction algorithm satisfy the linearity preserving criterion which requires that a discretization scheme should be exact on linear solutions. The vertex interpolation algorithm can be easily extended to 3 D case. Findings Numerical results show that it maintain optimal convergence rates for the solution and flux on 2 D and 3 D meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous. Originality/value This paper proposes a linearity preserving and explicit weighted vertex interpolation algorithm for cell-centered finite volume approximations of diffusion equations on general grids. The proposed finite volume scheme with the new interpolation algorithm allows arbitrary continuous or discontinuous diffusion tensors; the final scheme is applicable to arbitrary polygonal grids, which may have concave cells or degenerate ones with hanging nodes. The final scheme has second-order convergence rate for the approximate solution and higher than first-order accuracy for the flux on 2 D and 3 D meshes. The explicit weighted interpolation algorithm is easy to implement in three dimensions in case that the diffusion tensor is continuous or discontinuous.

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Section: Test 5: Nonlinear Problemmentioning

confidence: 89%

A linearity-preserving vertex interpolation algorithm for cell-centered finite volume approximations of anisotropic diffusion problems

Yang

Gao

2019

HFF

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“…These interpolation methods have been widely studied in different methods. 14,[23][24][25] However, there is a tremendous challenge in extending some 2D schemes for 3D schemes. It is not easy to design interpolation methods such as Gao 22 in 3D, mainly due to 3D mesh geometry and topology complexity.…”

Section: Introductionmentioning

confidence: 99%

“…One of the main features of the proposed interpolations, LPEW1 and LPEW2, which are constructed based on a flux conservative approach. These interpolation methods have been widely studied in different methods 14,23‐25 . However, there is a tremendous challenge in extending some 2D schemes for 3D schemes.…”

Section: Introductionmentioning

confidence: 99%

The positivity‐preserving finite volume scheme with fixed stencils for anisotropic diffusion problems on general polyhedral meshes

Yang

Gao

2022

Numerical Methods in Fluids

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Two kinds of nonlinear cell‐centered positivity‐preserving finite volume schemes are proposed for the anisotropic diffusion problems on general three‐dimensional polyhedral meshes. Firstly, the one‐sided flux on the cell‐faces is discretized using the fixed stencil of all vertices, then the cell‐centered discretization scheme is obtained using the nonlinear two‐point flux approximation. On this basis, a new explicit weighted second‐order vertex interpolation algorithm for arbitrary polyhedral meshes is designed to eliminate the vertex auxiliary unknowns in the scheme. In addition, an improved Anderson acceleration algorithm is adopted for nonlinear iteration. Finally, some benchmark examples are given to verify the convergence and positivity‐preserving property of the two schemes.

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“…A partir da malha primal (Triangularização de Delaunay), formada pelos triângulos esféricos é determinada a malha dual (Diagrama de Voronoi), formada pela união de todos os volumes de controle de Voronoi ou pela malha dual (Diagrama de Donald) (MacDonald et al, 1968;Su et al, 2018), formada pela união de todos os volumes de controle de Donald, conforme Figura 2.5. Os volumes de controle de Voronoi são obtidos ao se conectar os circuncentros de todos os triângulos esféricos adjacentes aos vértices da malha primal por segmentos geodésicos.…”

Section: Nível Vértices Triângulosunclassified

“…A malha esférica utilizada neste trabalho é a dual de Donald, formada por Diagramas de Donald (MacDonald et al, 1968;Su et al, 2018). Para cada volume de controle Ω i , e seus respectivos vizinhos, deve-se fazer a projeção dos vértices no plano tangente e tem-se uma outra malha formada por triângulos planos.…”

Section: Projeção Ortogonalunclassified

Um método robusto de volumes finitos de alta ordem para advecção em malhas esféricas geodésicas

Granjeiro¹

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A decoupled and positivity-preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes

Cited by 28 publications

References 50 publications

A linearity-preserving vertex interpolation algorithm for cell-centered finite volume approximations of anisotropic diffusion problems

A linearity-preserving vertex interpolation algorithm for cell-centered finite volume approximations of anisotropic diffusion problems

The positivity‐preserving finite volume scheme with fixed stencils for anisotropic diffusion problems on general polyhedral meshes

Um método robusto de volumes finitos de alta ordem para advecção em malhas esféricas geodésicas

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