2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Lingfa Kong ◽  
Yidao Dong ◽  
Wei Liu ◽  
Huaibao Zhang

AbstractAccuracy 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 [Kong, et al. Chin Phys B 29(10):100203, 2020], 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.


2018 ◽  
Vol 29 (12) ◽  
pp. 1850075
Author(s):  
Kotaro Kawai ◽  
Hông Vân Lê ◽  
Lorenz Schwachhöfer

In this paper, we show that a parallel differential form [Formula: see text] of even degree on a Riemannian manifold allows to define a natural differential both on [Formula: see text] and [Formula: see text], defined via the Frölicher–Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential with respect to the canonical parallel [Formula: see text]-form on a [Formula: see text]- and [Formula: see text]-manifold, respectively. We calculate the cohomology groups of [Formula: see text] and give a partial description of the cohomology of [Formula: see text].


1952 ◽  
Vol 4 ◽  
pp. 73-78
Author(s):  
Yûsaku Kawahara

In the book “Foundations of algebraic geometry” A. Weil proposed the following problem ; does every differential form of the first kind on a complete variety U determine on every subvariety V of U a differential form of the first kind? This problem was solved affirmatively by S. Koizumi when U is a complete variety without multiple point. In this note we answer this problem in affirmative in the case where V is a simple subvariety of a complete variety U (in §1). When the characteristic is 0 we may extend our result to a more general case but this does not hold for the case characteristic p≠0 (in §2).


Author(s):  
Loring W. Tu

This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ‎ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.


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