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 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.

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.

scholarly journals Comparison Study on the Performances of Finite Volume Method and Finite Difference Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Renwei Liu ◽  
Dongjie Wang ◽  
Xinyu Zhang ◽  
Wang Li ◽  
Bo Yu
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Difference Method ◽  
Convection Flow ◽  
Comparison Study ◽  
Square Cavity ◽  
Natural Convection Flow ◽  
Volume Method

Vorticity-stream function method and MAC algorithm are adopted to systemically compare the finite volume method (FVM) and finite difference method (FDM) in this paper. Two typical problems—lid-driven flow and natural convection flow in a square cavity—are taken as examples to compare and analyze the calculation performances of FVM and FDM with variant mesh densities, discrete forms, and treatments of boundary condition. It is indicated that FVM is superior to FDM from the perspective of accuracy, stability of convection term, robustness, and calculation efficiency. Particularly ,when the mesh is coarse and taken as20×20, the results of FDM suffer severe oscillation and even lose physical meaning.

scholarly journals Simulation of Earthquake Rupture Dynamics in Complex Geometries Using Coupled Finite Difference and Finite Volume Methods

2015 ◽  
Vol 17 (2) ◽  
pp. 337-370 ◽  
Author(s):  
Ossian O'Reilly ◽  
Jan Nordström ◽  
Jeremy E. Kozdon ◽  
Eric M. Dunham
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Difference Method ◽  
High Order ◽  
Complex Geometries ◽  
Earthquake Rupture ◽  
Volume Method ◽  
High Order Finite Difference

AbstractWe couple a node-centered finite volume method to a high order finite difference method to simulate dynamic earthquake ruptures along nonplanar faults in two dimensions. The finite volume method is implemented on an unstructured mesh, providing the ability to handle complex geometries. The geometric complexities are limited to a small portion of the overall domain and elsewhere the high order finite difference method is used, enhancing efficiency. Both the finite volume and finite difference methods are in summation-by-parts form. Interface conditions coupling the numerical solution across physical interfaces like faults, and computational ones between structured and unstructured meshes, are enforced weakly using the simultaneous-approximation-term technique. The fault interface condition, or friction law, provides a nonlinear relation between fields on the two sides of the fault, and allows for the particle velocity field to be discontinuous across it. Stability is proved by deriving energy estimates; stability, accuracy, and efficiency of the hybrid method are confirmed with several computational experiments. The capabilities of the method are demonstrated by simulating an earthquake rupture propagating along the margins of a volcanic plug.

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