颂歌
趋同(经济学)
数学
应用数学
稳健性(进化)
正多边形
稳态(化学)
缩放比例
数值分析
相场模型
领域(数学)
理论(学习稳定性)
准确度顺序
数值稳定性
计算机科学
相(物质)
数学分析
几何学
物理
生物化学
化学
物理化学
量子力学
纯数学
经济
基因
经济增长
机器学习
标识
DOI:10.1142/s0218202523500409
摘要
In this paper, we study the stability, accuracy and convergence behavior of various numerical schemes for phase-field modeling through a simple ODE model. Both theoretical analysis and numerical experiments are carried out on this ODE model to demonstrate the limitation of most numerical schemes that have been used in practice. One main conclusion is that the first-order fully implicit scheme is the only robust algorithm for phase-field simulations while all other schemes (that have been analyzed) may have convergence issue if the time step size is not exceedingly small. More specifically, by rigorous analysis in most cases, we have the following conclusions: (i) The first-order fully implicit scheme converges to the correct steady state solution for all time step sizes. In the case of multiple solutions, one of the solution branches always converges to the correct steady state solution. (ii) The first-order convex splitting scheme, which is equivalent to the first-order fully implicit scheme with a different time scaling, always converges to the correct steady state solution but may seriously lack numerical accuracy for transient solutions. (iii) For the second-order fully implicit and convex splitting schemes, for any time step size [Formula: see text], there exists an initial condition [Formula: see text], with [Formula: see text], such that the numerical solution converges to the wrong steady state solution. (iv) For [Formula: see text], all second-order schemes studied in this paper converge to the correct steady state solution although severe numerical oscillations occur for most of them if the time step size is not sufficiently small. (v) An unconditionally energy-stable scheme (such as the modified Crank–Nicolson scheme) is not necessarily better than a conditionally energy-stable scheme (such as the Crank–Nicolson scheme). Most, if not all, of the above conclusions are expected to be true for more general Allen–Cahn and other phase-field models.
科研通智能强力驱动
Strongly Powered by AbleSci AI