共查询到20条相似文献,搜索用时 31 毫秒
1.
三维多面体网格上扩散方程的保正格式 总被引:1,自引:0,他引:1
针对三维任意(星形)多面体网格, 本文构造了扩散方程的一种单元中心型非线性有限体积格式, 证明了该格式具有保正性. 在该格式设计中, 除引入网格中心量外, 还引入网格节点量和网格面中心量作为中间未知量, 它们将用网格中心未知量线性组合表示, 使得格式仅有网格中心未知量作为基本未知量. 在节点量计算中, 利用网格面上的调和平均点, 设计了一种适用于三维多面体网格的局部显式加权方法. 该格式适用于求解非平面的网格表面和间断扩散系数的问题. 数值例子验证了它对光滑解具有二阶精度和保正性. 相似文献
2.
3.
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 edge‐centered, 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. 相似文献
4.
Jiming Wu Zhiming Gao 《Applied Numerical Mathematics》2011,61(7):844-867
In this paper, for the structured quadrilateral mesh we derive a nine-point difference scheme which has five cell-centered unknowns and four vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear combination of the neighboring cell-centered unknowns, which reduces the scheme to a cell-centered one with a local stencil involving nine cell-centered unknowns. The coefficients in the linear combination are known as the weights and two types of new weights are proposed. These new weights are neither discontinuity dependent nor mesh topology dependent, have explicit expressions, can reduce to the one-dimensional harmonic-average weights on the nonuniform rectangular meshes, and moreover, are easily extended to the unstructured polygonal meshes and non-matching meshes. Both the derivation of the nine-point scheme and that of new weights satisfy the linearity preserving criterion. Numerical experiments show that, with these new weights, the nine-point difference scheme and its simple extension have a nearly second order accuracy on many highly distorted meshes, including structured quadrilateral meshes, unstructured polygonal meshes and non-matching meshes. 相似文献
5.
本文在星形多边形网格上, 构造了扩散方程新的单调有限体积格式.该格式与现有的基于非线性两点流的单调格式的主要区别是, 在网格边的法向流离散模板中包含当前边上的点, 在推导离散法向流的表达式时采用了定义于当前边上的辅助未知量, 这样既可适应网格几何大变形, 同时又兼顾了当前网格边上物理量的变化. 在光滑解情形证明了离散法向流的相容性.对于具有强各向异性、非均匀张量扩散系数的扩散方程, 证明了新格式是单调的, 即格式可以保持解析解的正性. 数值结果表明在扭曲网格上, 所构造的格式是局部守恒和保正的, 对光滑解有高于一阶的精度, 并且, 针对非平衡辐射限流扩散问题, 数值结果验证了新格式在计算效率和守恒精度上优于九点格式. 相似文献
6.
Calculating the vertex unknowns of nine point scheme on quadrilateral meshes for diffusion equation 总被引:1,自引:0,他引:1
In the construction of nine point scheme,both vertex unknowns and cell-centered unknowns are introduced,and the vertex unknowns are usually eliminated by using the interpolation of neighboring cell-centered unknowns,which often leads to lose accuracy.Instead of using interpolation,here we propose a different method of calculating the vertex unknowns of nine point scheme,which are solved independently on a new generated mesh.This new mesh is a Vorono¨i mesh based on the vertexes of primary mesh and some additional points on the interface.The advantage of this method is that it is particularly suitable for solving diffusion problems with discontinuous coeffcients on highly distorted meshes,and it leads to a symmetric positive definite matrix.We prove that the method has first-order convergence on distorted meshes.Numerical experiments show that the method obtains nearly second-order accuracy on distorted meshes. 相似文献
7.
Meihua Sheng Di Yang Zhiming Gao 《Numerical Methods for Partial Differential Equations》2023,39(2):1398-1424
In this paper, we propose a positivity-preserving conservative scheme based on the virtual element method (VEM) to solve convection–diffusion problems on general meshes. As an extension of finite element methods to general polygonal elements, the VEM has many advantages such as substantial mathematical foundations, simplicity in implementation. However, it is neither positivity-preserving nor locally conservative. The purpose of this article is to develop a new scheme, which has the same accuracy as the VEM and preserves the positivity of the numerical solution and local conservation on primary grids. The first step is to calculate the cell-vertex values by the lowest-order VEM. Then, the nonlinear two-point flux approximations are utilized to obtain the nonnegativity of cell-centered values and the local conservation property. The new scheme inherits both advantages of the VEM and the nonlinear two-point flux approximations. Numerical results show that the new scheme can reach the optimal convergence order of the virtual element theory, that is, the second-order accuracy for the solution and the first-order accuracy for its gradient. Moreover, the obtained cell-centered values are nonnegative, which demonstrates the positivity-preserving property of our new scheme. 相似文献
8.
In this paper, we present a new linear cell-centered finite volume multipoint flux approximation (MPFA-QL) scheme for discretizing diffusion problems on general polygonal meshes. This scheme uses a quasi-local stencil, based upon the conormal decomposition, to approximate the control face flux when solving the steady state diffusion problem, being able to reproduce piecewise linear solutions exactly and it is very robust when dealing with heterogeneous and highly anisotropic media and severely distorted meshes. In our linear scheme, we first construct the one-sided fluxes on each control surface independently and then a unique flux expression is obtained by a convex combination of the one-sided fluxes. The unknown values at the vertices that define a control surface are interpolated by means of a linearity-preserving interpolation procedure, considering control volumes surrounding these vertices. To show the potential of the MPFA-QL scheme, we solve some benchmark using triangular and quadrilateral meshes and we compare our scheme with other numerical formulations found in literature. 相似文献
9.
A parallel finite volume scheme preserving positivity for diffusion equation on distorted meshes 下载免费PDF全文
Zhiqiang Sheng Jingyan Yue Guangwei Yuan 《Numerical Methods for Partial Differential Equations》2017,33(6):2159-2178
Parallel domain decomposition methods are natural and efficient for solving the implicity schemes of diffusion equations on massive parallel computer systems. A finite volume scheme preserving positivity is essential for getting accurate numerical solutions of diffusion equations and ensuring the numerical solutions with physical meaning. We call their combination as a parallel finite volume scheme preserving positivity, and construct such a scheme for diffusion equation on distorted meshes. The basic procedure of constructing the parallel finite volume scheme is based on the domain decomposition method with the prediction‐correction technique at the interface of subdomains: First, we predict the values on each inner interface of subdomains partitioned by the domain decomposition. Second, we compute the values in each subdomain using a finite volume scheme preserving positivity. Third, we correct the values on each inner interface using the finite volume scheme preserving positivity. The resulting scheme has intrinsic parallelism, and needs only local communication among neighboring processors. Numerical results are presented to show the performance of our schemes, such as accuracy, stability, positivity, and parallel speedup.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2159–2178, 2017 相似文献
10.
Cheng Dong & Tong Kang 《高等学校计算数学学报(英文版)》2021,14(2):405-437
In this paper, we suggest a new vertex interpolation algorithm to improve
an existing cell-centered finite volume scheme for nonlinear diffusion problems on
general meshes. The new vertex interpolation algorithm is derived by applying a special limit procedure to the well-known MPFA-O method. Since the MPFA-O method
for 3D cases has been addressed in some studies, the new vertex interpolation algorithm can be extended to 3D cases naturally. More interesting is that the solvability
of the corresponding local system is proved under some assumptions. Additionally,
we modify the edge flux approximation by an edge-based discretization of diffusion coefficient, and thus the improved scheme is free of the so-called numerical
heat-barrier issue suffered by many existing cell-centered or hybrid schemes. The
final scheme allows arbitrary continuous or discontinuous diffusion coefficients and
can be applicable to arbitrary star-shaped polygonal meshes. A second-order convergence rate for the approximate solution and a first-order accuracy for the flux
are observed in numerical experiments. In the comparative experiments with some
existing vertex interpolation algorithms, the new algorithm shows obvious improvement on highly distorted meshes. 相似文献
11.
A cell edge‐based linearity‐preserving scheme for diffusion problems on two‐dimensional unstructured grids 下载免费PDF全文
Longshan Luo Zhiming Gao Jiming Wu 《Mathematical Methods in the Applied Sciences》2017,40(15):5423-5436
We propose a new finite volume scheme for 2D anisotropic diffusion problems on general unstructured meshes. The main feature lies in the introduction of two auxiliary unknowns on each cell edge, and then the scheme has both cell‐centered primary unknowns and cell edge‐based auxiliary unknowns. The auxiliary unknowns are interpolated by the multipoint flux approximation technique, which reduces the scheme to a completely cell‐centered one. The derivation of the scheme satisfies the linearity‐preserving criterion that requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is then called as a cell edge‐based linearity‐preserving scheme. The optimal convergence rates are numerically obtained on unstructured grids in case that the diffusion tensor is taken to be anisotropic and/or discontinuous. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
12.
Yulong Xing & Chi-Wang Shu 《数学研究》2014,47(3):221-249
In this paper, we survey our recent work on designing high order positivity-preserving
well-balanced finite difference and finite volume
WENO (weighted essentially non-oscillatory) schemes, and discontinuous Galerkin finite element
schemes for solving the shallow water equations with a non-flat bottom topography.
These schemes are genuinely high order accurate
in smooth regions for general solutions, are essentially non-oscillatory
for general solutions with discontinuities, and at the same time
they preserve exactly the water at rest or the more general moving water steady state solutions.
A simple positivity-preserving limiter, valid under suitable CFL condition,
has been introduced in one dimension and reformulated to two
dimensions with triangular meshes, and we prove that the resulting schemes guarantee
the positivity of the water depth. 相似文献
13.
Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multi-material system and explain the exchange of energy among electrons, ions and photons. Their highly nonlinear, strong discontinuous and tightly coupled phenomena always make the numerical solution of such equations extremely challenging. In this paper, we construct two finite volume element schemes both satisfying the discrete conservation property. One of them can well preserve the positivity of analytical solutions, while the other one does not satisfy this property. To fix this defect, two as repair techniques are designed. In addition, as the numerical simulation of 2-D 3-T equations is very time consuming, we also devise a mesh adaptation algorithm to reduce the cost. Numerical results show that these new methods are practical and efficient in solving this kind of problems. 相似文献
14.
In the past decades, the finite difference methods for space fractional operators develop rapidly; to the best of our knowledge, all the existing finite difference schemes, including the first and high order ones, just work on uniform meshes. The nonlocal property of space fractional operator makes it difficult to design the finite difference scheme on non-uniform meshes. This paper provides a basic strategy to derive the first and high order discretization schemes on non-uniform meshes for fractional operators. And the obtained first and second schemes on non-uniform meshes are used to solve space fractional diffusion equations. The error estimates and stability analysis are detailedly performed; and extensive numerical experiments confirm the theoretical analysis or verify the convergence orders. 相似文献
15.
We develop a Lions domain decomposition algorithm based on a cell functional minimization scheme on non-matching multi-block grids for nonlinear radiation diffusion equations, which are described by the coupled radiation diffusion equations of electron, ion and photon temperatures. The $L^2$orthogonal projection is applied in the Robin transmission condition of non-matching surfaces. Numerical results show that the algorithm keeps the optimal accuracy on the whole computational domain, is robust enough on distorted meshes and curved surfaces, and the convergence rate does not depend on Robin coefficients. It is a practical and attractive algorithm in applying to the two-dimensional three-temperature energy equations of Z-pinch implosion simulation. 相似文献
16.
本文对可压缩流体力学高精度拉格朗日格式及其保正性质近年来的发展给出回顾与综述.文中分别介绍了一维、二维可压缩流体力学方程中心型拉格朗日格式的设计步骤,回顾了高精度拉格朗日格式以及高精度保正拉格朗日格式的研究进展. 相似文献
17.
In this paper, we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral
grids that reduces to a cell-centered finite difference scheme. A special non-symmetric quadrature rule is employed that yields
a positive definite cell-centered system for the pressure by eliminating local velocities. The method is shown to be accurate
on highly distorted rough quadrilateral and hexahedral grids, including hexahedra with non-planar faces. Theoretical and numerical
results indicate first-order convergence for the pressure and face fluxes. 相似文献
18.
19.
Science China Mathematics - In this paper, a nonlinear finite volume scheme preserving the discrete maximum principle for the anisotropic diffusion equation on distorted meshes is described. We... 相似文献
20.
We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients
on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge
in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom
per element. A specially chosen quadrature rule for the L
2-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference
scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order
convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule
that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature
rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory. 相似文献