The SAV Scheme Based on the Barycentric Interpolation Collocation Method for the Allen-Cahn Equation
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摘要: 采用标量辅助变量(scalar auxiliary variable, SAV)方法结合重心插值配点法求解二维Allen-Cahn方程. 在时间方向上分别采用Crank-Nicolson格式、二阶向后差分格式离散，空间方向上采用重心Lagrange插值配点法离散，建立了两种无条件能量稳定SAV格式，并给出了重心插值配点格式的逼近性质. 数值实验表明：两种SAV配点格式的时间收敛阶为二阶，并满足能量递减规律. 与空间采用有限差分法离散对比，重心Lagrange配点格式具有指数收敛的特性.
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关键词:
- Allen-Cahn方程 /
- 重心Lagrange插值配点法 /
- SAV方法 /
- 能量稳定
Abstract: The scalar auxiliary variable (SAV) approach combined with the barycentric interpolation collocation method was proposed to solve the 2D Allen-Cahn equation. Two unconditional energy-stable SAV schemes were constructed based on the Crank-Nicolson scheme and the 2nd-order backward difference scheme for discretization in time, respectively, and the barycentric Lagrange interpolation collocation method for discretization in space. Moreover, the approximation properties of the barycentric Lagrange interpolation were presented. Numerical experiments show that the time-convergence rates of the 2 types of SAV schemes are of the 2nd order and both schemes satisfy the energy decay law. Compared with the finite difference method in space, the barycentric Lagrange interpolation collocation scheme features exponential convergence.-
Key words:
- Allen-Cahn equation /
- barycentric Lagrange interpolation collocation method /
- scalar auxiliary variable scheme /
- energy stability
1) 我刊编委赵景军推荐 -
图 1 不同数值格式的能量演化图(ε=0.1)
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 1. Time evolution curves of the free energy of different schemes(ε=0.1)
图 2 不同数值格式的空间收敛阶
Figure 2. The convergence rates in space of different numerical schemes
图 3 在不同时刻的数值快照(ε=0.06)
Figure 3. Snapshots of the numerical approximation at different moments(ε=0.06)
图 4 相变u在不同时刻的快照图(算例3)
Figure 4. Snapshots of the phase field u at different moments for example 3
图 5 u的数值解在不同时刻的快照(算例4)
Figure 5. Snapshots of the numerical approximation of u at different moments for example 4
表 1 采用CN2-BLI、BDF2-BLI格式求解u的L2误差
Table 1. The L2 errors of u solved by CN2-BLI and BDF2-BLI schemes
τ CN2-BLI order BDF2-BLI order 1.6×10-3 9.55×10-6 - 1.59×10-5 - 8×10-4 2.38×10-6 2.00 3.97×10-6 2.00 4×10-4 5.96×10-7 2.00 9.91×10-7 2.00 2×10-4 1.49×10-7 2.00 2.48×10-7 2.00 1×10-4 3.84×10-8 1.96 6.34×10-8 1.97 
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表 2 不同空间离散方案的精度对比结果
Table 2. The accuracy comparison of different discretization schemes in space
(M, N) CN2-BLI (M, N) BDF2-BLI (M, N) CN2-FD (8, 8) 3.25×10-4 (8, 8) 3.25×10-4 (40, 40) 2.54×10-3 (9, 9) 4.43×10-5 (9, 9) 4.43×10-5 (60, 60) 1.13×10-3 (10, 10) 6.57×10-6 (10, 10) 6.57×10-6 (80, 80) 6.34×10-4 (12, 12) 1.18×10-7 (12, 12) 1.09×10-7 (100, 100) 4.06×10-4 (15, 15) 2.98×10-8 (15, 15) 4.68×10-8 (120, 120) 2.82×10-4
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