Lack of robustness and accuracy of many numerical schemes for phase-field simulations,Mathematical Models and Methods in Applied Sciences

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Lack of robustness and accuracy of many numerical schemes for phase-field simulations
Mathematical Models and Methods in Applied Sciences ( IF 3.6 ) Pub Date : 2023-05-16 , DOI: 10.1142/s0218202523500409
Jinchao Xu _{1,

2} , Xiaofeng Xu ₂

Affiliation

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 $δ t > 0$ , there exists an initial condition $u_{0}$ , with $\| u_{0} \| > 1$ , such that the numerical solution converges to the wrong steady state solution.
(iv)	For $\| u_{0} \| \leq 1$ , 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.

中文翻译：

许多相场模拟数值方案缺乏鲁棒性和准确性

在本文中，我们通过简单的 ODE 模型研究了相场建模的各种数值方案的稳定性、准确性和收敛行为。对该 ODE 模型进行了理论分析和数值实验，以证明大多数已在实践中使用的数值格式的局限性。一个主要结论是，一阶完全隐式格式是相场模拟唯一鲁棒的算法，而如果时间步长不是非常小，所有其他格式（已经分析过）可能会出现收敛问题。更具体地说，通过对大多数情况的严格分析，我们得出以下结论：

（我）	对于所有时间步长，一阶完全隐式方案收敛到正确的稳态解。在有多个解的情况下，解分支之一总是收敛到正确的稳态解。
(二)	一阶凸分裂方案相当于具有不同时间尺度的一阶完全隐式方案，总是收敛到正确的稳态解，但可能严重缺乏瞬态解的数值精度。
(三)	对于二阶完全隐式和凸分裂方案，对于任何时间步长 $δ t > 0$ ，存在一个初始条件 $你_{0}$ ，和 $\| 你_{0} \| > 1$ ，使得数值解收敛到错误的稳态解。
（四）	为了 $\| 你_{0} \| \leq 1$ ，本文研究的所有二阶格式都会收敛到正确的稳态解，尽管如果时间步长不够小，大多数二阶格式都会出现严重的数值振荡。
（五）	无条件能量稳定方案（例如改进的克兰克-尼科尔森方案）不一定比有条件能量稳定方案（例如克兰克-尼科尔森方案）更好。