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Keywords = steady-state Allen-Cahn equation

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16 pages, 5440 KiB  
Article
A Modified Polynomial Expansion Algorithm for Solving the Steady-State Allen-Cahn Equation for Heat Transfer in Thin Films
by Chih-Wen Chang, Chein-Hung Liu and Cheng-Chi Wang
Appl. Sci. 2018, 8(6), 983; https://doi.org/10.3390/app8060983 - 15 Jun 2018
Cited by 8 | Viewed by 4529
Abstract
Meshfree algorithms offer a convenient way of solving nonlinear steady-state problems in arbitrary plane areas surrounded by complicated boundary shapes. The simplest of these is the polynomial expansion approach. However, it is rarely utilized as a primary tool for this purpose because of [...] Read more.
Meshfree algorithms offer a convenient way of solving nonlinear steady-state problems in arbitrary plane areas surrounded by complicated boundary shapes. The simplest of these is the polynomial expansion approach. However, it is rarely utilized as a primary tool for this purpose because of its rather ill-conditioned behavior. A well behaved polynomial expansion algorithm is presented in this paper which can be more effectively used to solve the steady-state Allen-Cahn (AC) equation for heat transfer in thin films. In this method, modified polynomial expansion was used to cope with each iteration of the steady-state Allen-Cahn equation to produce nonlinear algebraic equations where multiple scales are automatically determined by the collocation points. These scales can largely decrease the condition number of the coefficient matrix in each nonlinear system, so that the iteration process converges very quickly. The numerical solutions were found to be accurate and stable against moderate noise to better than 7.5%. Computational results verified the method and showed the steady-state Allen-Cahn equation for heat transfer in thin films could easily be resolved for several arbitrary plane domains. Full article
(This article belongs to the Special Issue Selected Papers from the 2017 International Conference on Inventions)
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