Volume 9, Issue 4
Effect of Nonuniform Grids on High-Order Finite Difference Method

Dan Xu, Xiaogang Deng, Yaming Chen, Guangxue Wang & Yidao Dong

Adv. Appl. Math. Mech., 9 (2017), pp. 1012-1034.

Published online: 2018-05

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  • Abstract

The finite difference (FD) method is popular in the computational fluid dynamics and widely used in various flow simulations. Most of the FD schemes are developed on the uniform Cartesian grids; however, the use of nonuniform or curvilinear grids is inevitable for adapting to the complex configurations and the coordinate transformation is usually adopted. Therefore the question that whether the characteristics of the numerical schemes evaluated on the uniform grids can be preserved on the nonuniform grids arises, which is seldom discussed. Based on the one-dimensional wave equation, this paper systematically studies the characteristics of the high-order FD schemes on nonuniform grids, including the order of accuracy, resolution characteristics and the numerical stability. Especially, the Fourier analysis involving the metrics is presented for the first time and the relation between the resolution of numerical schemes and the stretching ratio of grids is discussed. Analysis shows that for smooth varying grids, these characteristics can be generally preserved after the coordinate transformation. Numerical tests also validate our conclusions.

  • Keywords

Finite difference method, nonuniform grids, coordinate transformation, Fourier analysis.

  • AMS Subject Headings

65N06, 65N22

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-9-1012, author = {}, title = {Effect of Nonuniform Grids on High-Order Finite Difference Method}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {4}, pages = {1012--1034}, abstract = {

The finite difference (FD) method is popular in the computational fluid dynamics and widely used in various flow simulations. Most of the FD schemes are developed on the uniform Cartesian grids; however, the use of nonuniform or curvilinear grids is inevitable for adapting to the complex configurations and the coordinate transformation is usually adopted. Therefore the question that whether the characteristics of the numerical schemes evaluated on the uniform grids can be preserved on the nonuniform grids arises, which is seldom discussed. Based on the one-dimensional wave equation, this paper systematically studies the characteristics of the high-order FD schemes on nonuniform grids, including the order of accuracy, resolution characteristics and the numerical stability. Especially, the Fourier analysis involving the metrics is presented for the first time and the relation between the resolution of numerical schemes and the stretching ratio of grids is discussed. Analysis shows that for smooth varying grids, these characteristics can be generally preserved after the coordinate transformation. Numerical tests also validate our conclusions.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2016.m1477}, url = {http://global-sci.org/intro/article_detail/aamm/12187.html} }
TY - JOUR T1 - Effect of Nonuniform Grids on High-Order Finite Difference Method JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 1012 EP - 1034 PY - 2018 DA - 2018/05 SN - 9 DO - http://doi.org/10.4208/aamm.2016.m1477 UR - https://global-sci.org/intro/article_detail/aamm/12187.html KW - Finite difference method, nonuniform grids, coordinate transformation, Fourier analysis. AB -

The finite difference (FD) method is popular in the computational fluid dynamics and widely used in various flow simulations. Most of the FD schemes are developed on the uniform Cartesian grids; however, the use of nonuniform or curvilinear grids is inevitable for adapting to the complex configurations and the coordinate transformation is usually adopted. Therefore the question that whether the characteristics of the numerical schemes evaluated on the uniform grids can be preserved on the nonuniform grids arises, which is seldom discussed. Based on the one-dimensional wave equation, this paper systematically studies the characteristics of the high-order FD schemes on nonuniform grids, including the order of accuracy, resolution characteristics and the numerical stability. Especially, the Fourier analysis involving the metrics is presented for the first time and the relation between the resolution of numerical schemes and the stretching ratio of grids is discussed. Analysis shows that for smooth varying grids, these characteristics can be generally preserved after the coordinate transformation. Numerical tests also validate our conclusions.

Dan Xu, Xiaogang Deng, Yaming Chen, Guangxue Wang & Yidao Dong. (2020). Effect of Nonuniform Grids on High-Order Finite Difference Method. Advances in Applied Mathematics and Mechanics. 9 (4). 1012-1034. doi:10.4208/aamm.2016.m1477
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