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 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.
Highlights
Much interest has been shown in the solution of the Allen-Cahn (AC) equation, the primary purpose of which has been depiction of the motion of antiphase boundaries in crystalline solids and phase separation in binary alloys [1]
The main reason being that the resultant nonlinear algebraic equations (NAEs) are often seriously ill-conditioned
To acquire an accurate solution of an AC equation employing the modified Pascal triangle polynomial expansion scheme, we have to develop a more accurate and effective method to cope with these NAEs by reducing the condition numbers
Summary
Much interest has been shown in the solution of the Allen-Cahn (AC) equation, the primary purpose of which has been depiction of the motion of antiphase boundaries in crystalline solids and phase separation in binary alloys [1]. Bulent et al [15] investigated the numerical solution of the AC equation with constant and degenerate mobility, as well as with polynomial and logarithmic energy functionals They discretized the model equation using the symmetric interior penalty Galerkin method in space, and by the average vector field (AVF) method in time. We employ the polynomial expansion as a trial solution of the PDE and formulate the needed algebraic equations after a suitable collocation in the problem area This is rarely utilized as a major numerical tool to resolve the nonlinear PDEs. The main reason being that the resultant nonlinear algebraic equations (NAEs) are often seriously ill-conditioned. To acquire an accurate solution of an AC equation employing the modified Pascal triangle polynomial expansion scheme, we have to develop a more accurate and effective method to cope with these NAEs by reducing the condition numbers.
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