scholarly journals Resolution of the Generalized Eigenvalue Problem in the Neutron Diffusion Equation Discretized by the Finite Volume Method

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Álvaro Bernal ◽  
Rafael Miró ◽  
Damián Ginestar ◽  
Gumersindo Verdú
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Eigenvalue Problem ◽  
Diffusion Equation ◽  
Finite Volume Method ◽  
Difference Method ◽  
Neutron Diffusion ◽  
Generalized Eigenvalue ◽  
Volume Method ◽  
Neutron Diffusion Equation

Numerical methods are usually required to solve the neutron diffusion equation applied to nuclear reactors due to its heterogeneous nature. The most popular numerical techniques are the Finite Difference Method (FDM), the Coarse Mesh Finite Difference Method (CFMD), the Nodal Expansion Method (NEM), and the Nodal Collocation Method (NCM), used virtually in all neutronic diffusion codes, which give accurate results in structured meshes. However, the application of these methods in unstructured meshes to deal with complex geometries is not straightforward and it may cause problems of stability and convergence of the solution. By contrast, the Finite Element Method (FEM) and the Finite Volume Method (FVM) are easily applied to unstructured meshes. On the one hand, the FEM can be accurate for smoothly varying functions. On the other hand, the FVM is typically used in the transport equations due to the conservation of the transported quantity within the volume. In this paper, the FVM algorithm implemented in the ARB Partial Differential Equations solver has been used to discretize the neutron diffusion equation to obtain the matrices of the generalized eigenvalue problem, which has been solved by means of the SLEPc library.

scholarly journals Discontinuity Capture in One-Dimensional Space Using the Numerical Manifold Method with High-Order Legendre Polynomials

2020 ◽  
Vol 10 (24) ◽  
pp. 9123
Author(s):  
Yan Zeng ◽  
Hong Zheng ◽  
Chunguang Li
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Finite Volume Method ◽  
Difference Method ◽  
Legendre Polynomials ◽  
Dimensional Space ◽  
Numerical Manifold Method ◽  
One Dimensional ◽  
Numerical Manifold ◽  
Volume Method

Traditional methods such as the finite difference method, the finite element method, and the finite volume method are all based on continuous interpolation. In general, if discontinuity occurred, the calculation result would show low accuracy and poor stability. In this paper, the numerical manifold method is used to capture numerical discontinuities, in a one-dimensional space. It is verified that the high-degree Legendre polynomials can be selected as the local approximation without leading to linear dependency, a notorious “nail” issue in Numerical Manifold Method. A series of numerical tests are carried out to evaluate the performance of the proposed method, suggesting that the accuracy by the numerical manifold method is higher than that by the later finite difference method and finite volume method using the same number of unknowns.

scholarly journals A new formulation of finite difference and finite volume methods for solving a space fractional convection–diffusion model with fewer error estimates

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Reem Edwan ◽  
Shrideh Al-Omari ◽  
Mohammed Al-Smadi ◽  
Shaher Momani ◽  
Andreea Fulga
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Equation ◽  
Finite Volume ◽  
Difference Method ◽  
Order Accuracy ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
The Finite Difference Method ◽  
Volume Method

AbstractConvection and diffusion are two harmonious physical processes that transfer particles and physical quantities. This paper deals with a new aspect of solving the convection–diffusion equation in fractional order using the finite volume method and the finite difference method. In this context, we present an alternative way for estimating the space fractional derivative by utilizing the fractional Grünwald formula. The proposed methods are conditionally stable with second-order accuracy in space and first-order accuracy in time. Many comparisons are performed to display reliability and capability of the proposed methods. Furthermore, several results and conclusions are provided to indicate appropriateness of the finite volume method in solving the space fractional convection–diffusion equation compared with the finite difference method.

Research on Nodal Expansion Method for Transient Convection Diffusion Equation

2014 ◽  
Author(s):  
Xiafeng Zhou ◽  
Fu Li
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Equation ◽  
Expansion Method ◽  
Neutron Diffusion ◽  
Dimensional Problem ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Nodal Expansion Method ◽  
Neutron Diffusion Equation

Nodal expansion method (NEM), well known for its high accuracy and efficiency, has been widely applied to reactor physics analysis. It is proven that NEM has an advantage over traditional finite difference method (FDM) and finite volume method (FVM). However, for most reactor thermal hydraulic codes, traditional FDM or FVM is still in use, and the NEM is barely utilized. Therefore, to make full use of the advantages of NEM and effectively solve the thermal hydraulic problems, the derivation and analytical process of nodal expansion method for transient convection-diffusion equation is studied in this paper. First, time discretization is derived by finite difference method, and then is manipulated to ensure that the form of convection-diffusion equation is consistent with that of neutron diffusion equation. After that, the approach of NEM for neutron diffusion equation can be easily utilized in the thermal hydraulic codes, and the code TNEM based on NEM is developed to solve the multi-dimensional transient convection-diffusion equation. At last, through the numerical benchmarks and error analysis, the numerical results of TNEM are found to agree well with the reference solutions and are superior to that of center difference scheme and first order upwind scheme as for the one-dimensional problem and multi-dimensional problem. Furthermore, good accuracy can be maintained even for coarse meshes.

Comparison of Finite Difference and Finite Volume Method for Numerical Simulation of the Incompressible Viscous Driven Cavity Flow

2013 ◽  
Vol 732-733 ◽  
pp. 413-416
Author(s):  
Jian Wang ◽  
Jiang Fei Li ◽  
Wen Xue Cheng ◽  
Lian Yuan ◽  
Bo Li ◽  
...  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Difference Method ◽  
Cavity Flow ◽  
Second Order ◽  
Central Difference ◽  
Driven Cavity ◽  
Volume Method

In this paper, finite difference method and finite volume method are applied to incompressible viscous driven cavity flow problems, and their results are analyzed and compared. As for the finite difference method, second-order upwind and second-order central difference format are applied to the discretization of the convection and diffusion items respectively. For the finite volume method, three different ways are utilized to discretize the control equations: QUICK, second-order central difference and third-order upwind formats. The results show that computing time as well as calculation accuracy exponentially depends on Reynolds number, discrete formats and grid numbers.

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