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We construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes. 相似文献
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In this paper, we propose a new positivity-preserving finite volume scheme with fixed stencils for the nonequilibrium radiation diffusion equations on distorted meshes. This scheme is used to simulate the equations on meshes with both the cell-centered and cell-vertex unknowns. The cell-centered unknowns are the primary unknowns, and the element vertex unknowns are taken as the auxiliary unknowns, which can be calculated by interpolation algorithm. With the nonlinear two-point flux approximation, the interpolation algorithm is not required to be positivity-preserving. Besides, the scheme has a fixed stencil and is locally conservative. The Anderson acceleration is used for the Picard method to solve the nonlinear systems efficiently. Several numerical results are also given to illustrate the efficiency and strong positivity-preserving quality of the scheme. 相似文献
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Jiming Wu Zihuan Dai Zhiming Gao Guangwei Yuan 《Journal of computational physics》2010,229(9):3382-3401
In this paper, we employ the so-called linearity preserving method, which requires that a difference scheme should be exact on linear solutions, to derive a nine-point difference scheme for the numerical solution of diffusion equation on the structured quadrilateral meshes. This scheme uses firstly both cell-centered unknowns and vertex unknowns, and then the vertex unknowns are treated as a linear combination of the surrounding cell-centered unknowns, which reduces the scheme to a cell-centered one. The weights in the linear combination are derived through the linearity preserving approach and can be obtained by solving a local linear system whose solvability is rigorously discussed. Moreover, the relations between our linearity preserving scheme and some existing schemes are also discussed, by which a generalized multipoint flux approximation scheme based on the linearity preserving criterion is suggested. Numerical experiments show that the linearity preserving schemes in this paper have nearly second order accuracy on many highly skewed and highly distorted structured quadrilateral meshes. 相似文献
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多流管方法是二维多介质辐射流体力学数值模拟中一类常用的求解方法,它采用Lagrange-Euler混合型四边形网格,称为多流管网格.通常其网格品质高于一般的四边形网格.在这类网格上,可以利用网格特性对九点扩散格式中的节点插值方法进行改进.本文利用调和平均点和梯度离散构造的方法提出几种节点插值方法.并给出数值实验,说明现... 相似文献
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A new reconstruction algorithm is proposed for constructing cell-centered diffusion schemes on distorted meshes. Its main feature is that edge unknowns are defined at certain balance points, the locations of which depend on the diffusion coefficient and the skewness of grid cells, so as to obtain a two-point reconstruction stencil. Implementing the new algorithm for the approximation of gradients, we extend the IDC (improved deferred correction) scheme, which was proposed by Traoré et al. [P. Traoré, Y. Ahipo, C. Louste, A robust and efficient finite volume scheme for the discretization of diffusive flux on extremely skewed meshes in complex geometries, J. Comput. Phys. 228 (2009) 5148–5159], to handle diffusion problems with discontinuous coefficients. Numerical results demonstrate the accuracy and efficiency of the extended scheme. 相似文献
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《Journal of computational physics》2008,227(1):492-512
We consider a non-linear finite volume (FV) scheme for stationary diffusion equation. We prove that the scheme is monotone, i.e. it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients. The scheme is extended to regular star-shaped polygonal meshes and isotropic heterogeneous coefficients. 相似文献
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We developed a new monotone finite volume method for diffusion equations. The second-order linear methods, such as the multipoint flux approximation, mixed finite element and mimetic finite difference methods, are not monotone on strongly anisotropic meshes or for diffusion problems with strongly anisotropic coefficients. The finite volume (FV) method with linear two-point flux approximation is monotone but not even first-order accurate in these cases. The developed monotone method is based on a nonlinear two-point flux approximation. It does not require any interpolation scheme and thus differs from other nonlinear finite volume methods based on a two-point flux approximation. The second-order convergence rate is verified with numerical experiments. 相似文献
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We are interested in a robust and accurate finite volume scheme for 2-D parabolic problems derived from the cell functional minimization approach. The scheme has a local stencil, is locally conservative, treats discontinuity rigorously and leads to a symmetric positive definite linear system. Since the scheme has both cell centered unknowns and cell edge unknowns, the computational cost is an issue and a parallel algorithm is then suggested based on nonoverlapping domain decomposition approach. The interface condition is of the Dirichlet–Robin type and has a parameter λ. By choosing this parameter properly, the convergence of the iteration process could be sped up. Numerical results for linear and nonlinear problems demonstrate the good performance of the cell functional minimization scheme and its parallel version on distorted meshes. 相似文献
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We present a new collocated numerical scheme for the approximation of the Navier–Stokes and energy equations under the Boussinesq assumption for general grids, using the velocity–pressure unknowns. This scheme is based on a recent scheme for the diffusion terms. Stability properties are drawn from particular choices for the pressure gradient and the non-linear terms. Convergence of the approximate solutions may be proven mathematically. Numerical results show the accuracy of the scheme on irregular grids. 相似文献
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G. Capdeville 《Journal of computational physics》2008,227(5):2977-3014
This paper proposes a new WENO procedure to compute multi-scale problems with embedded discontinuities, on non-uniform meshes.In a one-dimensional context, the WENO procedure is first defined on a five-points stencil and designed to be fifth-order accurate in regions of smoothness. To this end, we define a finite-volume discretization in which we consider the cell averages of the variable as the discrete unknowns. The reconstruction of their point-values is then ensured by a unique fifth-order polynomial. This optimum polynomial is considered as a symmetric and convex combination, by ideal weights, of four quadratic polynomials.The symmetric nature of the resulting interpolation has an important consequence: the choice of ideal weights has no influence on the accuracy of the discretization. This advantage enables to formulate the interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights.We adapt this procedure for the non-linear weights to maintain the theoretical convergence properties of the optimum reconstruction, whatever the problem considered.The resulting scheme is a fifth-order WENO method based on central interpolation and TVD Runge–Kutta time-integration. We call this scheme the CWENO5 scheme.Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In those experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems. Finally, the new algorithm is directly extended to bi-dimensional problems. 相似文献
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Extending Seventh-Order Dissipative Compact Scheme Satisfying Geometric Conservation Law to Large Eddy Simulation on Curvilinear Grids
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Yi Jiang Meiliang Mao Xiaogang Deng & Huayong Liu 《advances in applied mathematics and mechanics.》2015,7(4):407-429
Seventh-order hybrid cell-edge and cell-node dissipative compact scheme
(HDCS-E8T7) is extended to a new implicit large eddy simulation named HILES on
stretched and curvilinear meshes. Although the conception of HILES is similar to that
of monotone integrated LES (MILES), i.e., truncation error of the discretization scheme
itself is employed to model the effects of unresolved scales, HDCS-E8T7 is a new high-order
finite difference scheme, which can eliminate the surface conservation law (SCL)
errors and has inherent dissipation. The capability of HILES is tested by solving several
benchmark cases. In the case of flow past a circular cylinder, the solutions of HILES
fulfilling the SCL have good agreement with the corresponding experiment data, however,
the flow field is gradually contaminated when the SCL error is enlarged. With the
help of fulling the SCL, ability of HILES for handling complex geometry has been enhanced.
The numerical solutions of flow over delta wing demonstrate the potential of
HILES in simulating turbulent flow on complex configuration. 相似文献
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讨论抛物型方程的离散差分格式的构造,对九点差分格式进行了适用范围的讨论,并在此基础上提出辅助网格差分方法,用于处理因网格长宽比大且扭曲较大的网格引起的计算精度与计算效率降低的问题,该方法从守恒方程出发,将九点差分格式应用于按某种合适的方式进行重分之后的网格上,减少由于网格正则性差以及网格节点上的物理量采用周围网格量的加权平均等原因所引起的计算误差,得到一个新的但其解仍然逼近原来网格上的物理量的方程组.所构造的方法便于实施,且更适合于对实际物理模型的模拟,能比较好地适应流体大变形导致的网格扭曲,数值试验表明它有较好的数值精度和稳定性. 相似文献
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通过构造新的平衡分布函数和结合分区自适应网格加密方法,对不带扩散项的平衡辐射流体力学方程,构造二阶的分子动理学BGK-AMR格式.一方面在关心的计算区域中局部加密计算网格,提高计算精度的同时大大节省了计算网格数量和计算时间;另一方面,不同于已有的参数强耦合平衡分布函数,新构造的平衡分布函数中各参数不相互依赖,简化了辐射流体力学分子动理学格式的计算.一维和二维的数值算例显示了格式的性能. 相似文献
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Pierre-Henri Maire 《Journal of computational physics》2009,228(7):2391-2425
We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme. 相似文献
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任意网格重映的样条逼近算法 总被引:2,自引:1,他引:1
在大变形流体力学问题的数值模拟中,任何方法都必须考虑网格重分或网格自适应,只要改动网格就涉及重分,或自适应后从旧的、扭曲的网格到新网格的守恒量重映,包括质量、动量和能量.在研究样条函数逼近的基础上,给出一种物理量重映的对结构网格和非结构网格均适应的算法,并给出了数值结果. 相似文献
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基于非结构四边形网格发展求解双曲守恒律的三阶加权基本无振荡(WENO)格式.针对任意非结构四边形网格选取重构模板,并给出基于线性多项式的三阶线性重构.但对于一般的非结构四边形网格,会出现非常大的线性权和负权,使得非线性重构的WENO格式对光滑问题也不稳定.本文给出一个处理非常大的线性权的优化重构方法,对优化后得到的负线性权采用分裂方法进行处理.对于非线性权,提出一种考虑局部网格和物理量间断的新光滑度量因子.采用优化重构方法和新的非线性权,当前的三阶WENO格式在质量很差的网格上也具有很好的稳定性.理论的三阶精度在数值精度测试算例中得到验证,同时一范数和无穷范数的误差绝对值不依赖于网格质量;具有强间断的数值结果证明了当前格式的有效性. 相似文献