Effects of high-frequency damping on iterative convergence of implicit viscous solver

Nishikawa, Hiroaki; Nakashima, Yoshitaka; Watanabe, Norihiko

doi:10.1016/j.jcp.2017.07.021

Cited by 34 publications

(13 citation statements)

References 19 publications

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“…The gradients ∇ a are used in an auxiliary fashion, in order to make the diffusive fluxes consistent. However, it has recently been shown [43][44][45] that schemes such as (48) and (49) can also be interpreted as equivalent to substituting an interpolated value of ∇ a φ for ∇φ(c f ) in (47) and adding a damping term which is a function of the direct difference φ(N f ) − φ(P ) and of the gradients ∇ a φ(P ) and ∇ a φ(N f ) and tends to zero with grid refinement.…”

Section: Tests With An In-house Solvermentioning

confidence: 99%

A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

et al. 2017

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Finite volume methods (FVMs) constitute a popular class of methods for the numerical simulation of fluid flows. Among the various components of these methods, the discretisation of the gradient operator has received less attention despite its fundamental importance with regards to the accuracy of the FVM. The most popular gradient schemes are the divergence theorem (DT) (or Green-Gauss) scheme, and the least-squares (LS) scheme. Both are widely believed to be secondorder accurate, but the present study shows that in fact the common variant of the DT gradient is second-order accurate only on structured meshes whereas it is zeroth-order accurate on general unstructured meshes, and the LS gradient is second-order and first-order accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. The schemes are then used within a FVM to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zeroth-order accuracy of the DT gradient is inherited by the FVM as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the LS gradient leads to second-order accurate results, as does the use of alternative, consistent, DT gradient schemes, including a new iterative scheme that makes the common DT gradient consistent at almost no extra cost. The numerical tests are performed using both an in-house code and the popular public domain PDE solver OpenFOAM. This is the accepted version of the article published in:

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Section: Tests With An In-house Solvermentioning

confidence: 99%

A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

et al. 2017

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“…Here are many different schemes for adding the jump term, such as the edge-normal (EN) and face-tangent (FT) schemes [50]. Based on the FT scheme, a novel α-damping scheme is proposed by Nishikawa [51][52][53][54]. The accuracy as well as robustness of second-order unstructured finite volume solver is greatly improved by this scheme, and it could be formulated as (44) where,  is a damping coefficient, and here, a special value is 43  = , which has been known to provide superior accuracy and robustness [28,48,51].…”

Section: Dissipative Term Evaluation (Governed By Laplacian Equation)mentioning

confidence: 99%

Extending the global-direction stencil with "face-area-weighted centroid" to unstructured finite volume discretization from integral form

Kong

Dong²,

Liu

et al. 2020

Preprint

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Accuracy of unstructured finite volume discretization is greatly influenced by the gradient reconstruction. For the commonly used k-exact reconstruction method, the cell centroid is always chosen as the reference point to formulate the reconstructed function. But in some practical problems, such as the boundary layer, cells in this area are always set with high aspect ratio to improve the local field resolution, and if geometric centroid is still utilized for the spatial discretization, the severe grid skewness cannot be avoided, which is adverse to the numerical performance of unstructured finite volume solver. In previous work [Chinese Physics B. 2020, In press], we explored a novel global-direction stencil and combined it with the face-area-weighted centroid on unstructured finite volume methods from differential form to realize the skewness reduction and a better reflection of flow anisotropy. Greatly inspired by the differential form, in this research, we demonstrate that it is also feasible to extend this novel method to the unstructured finite volume discretization from integral form on both second and third-order finite volume solver. Numerical examples governed by linear convective, Euler and Laplacian equations are utilized to examine the correctness as well as effectiveness of this extension. Compared with traditional vertex-neighbor and face-neighbor stencils based on the geometric centroid, the grid skewness is almost eliminated and computational accuracy as well as convergence rate is greatly improved by the global-direction stencil with face-area-weighted centroid. As a result, on unstructured finite volume discretization from integral form, the method also has superiorities on both computational accuracy and convergence rate.

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Section: Supersonic Vortex Flow (Governed By Euler Equations)mentioning

confidence: 99%

Extending the global-direction stencil with "face-area-weighted centroid" to unstructured finite volume discretization from integral form

Kong

Dong

Liu

et al. 2020

Preprint

View full text Add to dashboard Cite

Accuracy of unstructured finite volume discretization is greatly influenced by the gradient reconstruction. For the commonly used k-exact reconstruction method, the cell centroid is always chosen as the reference point to formulate the reconstructed function. But in some practical problems, such as the boundary layer, cells in this area are always set with high aspect ratio to improve the local field resolution, and if geometric centroid is still utilized for the spatial discretization, the severe grid skewness cannot be avoided, which is adverse to the numerical performance of unstructured finite volume solver. In previous work, we explored a novel global-direction stencil and combine it with face-area-weighted centroid on unstructured finite volume methods from differential form to realize the skewness reduction and a better reflection of flow anisotropy. Note, however, that the differential form is hard to achieve higher-order accuracy, and in order to set stage for the method promotion on higher-order numerical simulation, in this research, we demonstrate that it is also feasible to extend this novel method to the unstructured finite volume discretization in integral form. Numerical examples governed by linear convective, Euler and Laplacian equations are utilized to examine the correctness as well as effectiveness of this extension. Compared with traditional vertex-neighbor and face-neighbor stencils based on the geometric centroid, the grid skewness is almost eliminated and computational accuracy as well as convergence rate is greatly improved by the global-direction stencil with face-area-weighted centroid. As a result, on unstructured finite volume discretization from integral form, the method also has a better numerical performance.

show abstract

Effects of high-frequency damping on iterative convergence of implicit viscous solver

Cited by 34 publications

References 19 publications

A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

Extending the global-direction stencil with "face-area-weighted centroid" to unstructured finite volume discretization from integral form

Extending the global-direction stencil with "face-area-weighted centroid" to unstructured finite volume discretization from integral form

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