A Third Order Accurate in Time, BDF-Type Energy Stable Scheme for the Cahn-Hilliard Equation
A Third Order Accurate in Time, BDF-Type Energy Stable Scheme for the Cahn-Hilliard Equation
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DOI：
10.4208/nmtma.oa-2021-0165
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发表时间：
2022-06
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影响因子：
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通讯作者：
Kelong Cheng;Cheng Wang;S. Null;Yanmei Wu
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文献类型：
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作者：
Kelong Cheng;Cheng Wang;S. Null;Yanmei Wu
在本文中，我们针对具有三阶时间精度的Cahn - Hilliard方程提出并分析了一种向后差分公式（BDF）型数值格式。采用傅里叶伪谱方法对空间进行离散。表面扩散项和非线性化学势项采用隐式处理，而出于可解性的考虑，膨胀项由一个三阶显式外推公式来近似。此外，在数值格式中添加了一个形如\(-A_0\Delta t^2\Delta_N(\varphi^{n + 1}-\varphi^n)\)的三阶精确的Douglas - Dupont正则化项。特别地，在一个修正版本中仔细推导了能量稳定性，从而得到原始能量泛函的一个一致有界性，并且系数\(A\)有了理论依据。作为这种能量稳定性分析的结果，得到了数值解的一个时间一致的\(L^6_N\)有界性。并且，借助数值解的\(L^6_N\)有界性，在\(L^\infty_{\Delta t}(0,T;L^2_N)\cap L^{2}_{\Delta t}(0,T;H^2_h)\)范数下给出了最优速率收敛分析和误差估计。给出了一些数值模拟结果以证明数值格式的有效性和三阶收敛性。
. In this paper we propose and analyze a backward differentiation formula (BDF) type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy. The Fourier pseudo-spectral method is used to discretize space. The surface diffusion and the nonlinear chemical potential terms are treated implicitly, while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont regularization term, in the form of − A 0 ∆ t 2 ∆ N ( φ n +1 − φ n ) , is added in the numerical scheme. In particular, the energy stability is carefully derived in a modified version, so that a uniform bound for the original energy functional is available, and a theoretical justification of the coefficient A becomes available. As a result of this energy stability analysis, a uniform-in-time L 6 N bound of the numerical solution is obtained. And also, the optimal rate convergence analysis and error estimate are provided, in the L ∞ ∆ t (0 , T ; L 2 N ) ∩ L 2∆ t (0 , T ; H 2 h ) norm, with the help of the L 6 N bound for the numerical solution. A few numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.