Finite analytic numerical method for solving two‐dimensional quasi‐Laplace equation |
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Authors: | Zheng‐Xian Qu Zhi‐Feng Liu Xiao‐Hong Wang Peng Zhao |
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Affiliation: | 1. Department of Thermal Science and Energy Engineering, University of Science and Technology of China, , Hefei, Anhui, 230026 People's Republic of China;2. Department of Mechanical and Aerospace Engineering, Princeton University, , Princeton, New Jersey, 08544 USA |
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Abstract: | A typical power series analytic solution of quasi‐Laplace equation in the infinitesimal angle domain around the singular point of the square cells is provided in this article. Toward the singular point, the gradient of the potential variable will tend to infinity, which is described by the first term of the power series solution. Based on this analytic solution, three finite analytic numerical methods are proposed. These methods are analogous and are constructed, respectively, when considering different numbers of the terms or using different schemes to determine the relevant parameters in the power series. Numerical examples show that all of the three finite analytic numerical methods proposed can provide rather accurate solutions than the traditional numerical methods. In contrast, when using the traditional numerical schemes to solve the quasi‐Laplace equation in a strong heterogeneous medium, the refinement ratio for the grid cell needs to increase dramatically to get an accurate result. In practical applications, subdividing each origin cell into 2 × 2 or 3 × 3 subcells is enough for the finite analytical numerical methods to get relatively accurate results. The finite analytical numerical methods are also convenient to construct the flux field with high accuracy.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1755–1769, 2014 |
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Keywords: | finite analytic method quasi‐Laplace equation heterogeneous media |
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