计算物理 ›› 2022, Vol. 39 ›› Issue (6): 641-650.DOI: 10.19596/j.cnki.1001-246x.8515
赵国忠1(
), 蔚喜军2, 董自明1, 郭虹平1, 郭鹏云1, 李姝敏1
收稿日期:2022-02-17
出版日期:2022-11-25
发布日期:2023-04-01
作者简介:Zhao Guozhong (1977-), male, PhD, professor, research in computational fluid dynamics, E-mail: zhaoguozhongbttc@sina.com
Guozhong ZHAO1(), Xijun YU2, Ziming DONG1, Hongping GUO1, Pengyun GUO1, Shumin LI1
Received:2022-02-17
Online:2022-11-25
Published:2023-04-01
摘要：
构造一类求解非线性薛定谔方程的局部间断Petrov-Galerkin方法。利用构造的方法模拟几种类型的孤立子并讨论与孤立子密切相关的一些现象, 包括孤立子的传播与碰撞, 动孤立子和驻孤立子的生成, N孤立子的有界态。该方法可以模拟孤立子相关现象中一些复杂结构。数值实验表明该方法具有高阶精度且可以达到最优收敛阶。局部间断Petrov-Galerkin方法的计算效率与局部间断Galerkin方法相当, 但计算公式简单。
赵国忠, 蔚喜军, 董自明, 郭虹平, 郭鹏云, 李姝敏. 非线性Schrödinger方程几类孤立子解: 局部间断Petrov-Galerkin方法[J]. 计算物理, 2022, 39(6): 641-650.
Guozhong ZHAO, Xijun YU, Ziming DONG, Hongping GUO, Pengyun GUO, Shumin LI. Several Kinds of Soliton Solution of Nonlinear Schrödinger Equation: Local Discontinuous Petrov-Galerkin Method[J]. Chinese Journal of Computational Physics, 2022, 39(6): 641-650.
| pk | grid | ‖eu‖L1 | Order | ‖eu‖L2 | Order | ‖eu‖L∞ | Order |
| p1 | 20 | 6.358 7×10-2 | 2.843 0×10-2 | 1.411 7×10-2 | |||
| 40 | 1.594 3×10-2 | 1.995 8 | 7.121 3×10-3 | 1.997 2 | 3.633 0×10-3 | 1.958 2 | |
| 80 | 3.999 5×10-3 | 1.995 0 | 1.785 3×10-3 | 1.995 9 | 9.215 8×10-4 | 1.979 0 | |
| 160 | 1.002 1×10-3 | 1.996 8 | 4.471 8×10-4 | 1.997 3 | 2.320 7×10-4 | 1.989 5 | |
| p2 | 20 | 7.591 4×10-4 | 4.373 1×10-4 | 2.204 2×10-4 | |||
| 40 | 9.387 9×10-5 | 3.015 5 | 5.404 4×10-5 | 3.016 5 | 2.734 7×10-5 | 3.010 8 | |
| 80 | 1.156 0×10-5 | 3.021 7 | 6.666 6×10-6 | 3.019 1 | 3.367 5×10-6 | 3.021 6 | |
| 160 | 1.444 0×10-6 | 3.001 0 | 8.328 3×10-7 | 3.000 9 | 4.206 3×10-7 | 3.001 1 |
Table 1 L1, L2 and L∞ errors for real part of u at time 1.0 in [0, 2π] with α = 0.5, β = γ = 0, A = c = 1 in Example 1 by using linear and quadratic elements
| pk | grid | ‖eu‖L1 | Order | ‖eu‖L2 | Order | ‖eu‖L∞ | Order |
| p1 | 20 | 6.358 7×10-2 | 2.843 0×10-2 | 1.411 7×10-2 | |||
| 40 | 1.594 3×10-2 | 1.995 8 | 7.121 3×10-3 | 1.997 2 | 3.633 0×10-3 | 1.958 2 | |
| 80 | 3.999 5×10-3 | 1.995 0 | 1.785 3×10-3 | 1.995 9 | 9.215 8×10-4 | 1.979 0 | |
| 160 | 1.002 1×10-3 | 1.996 8 | 4.471 8×10-4 | 1.997 3 | 2.320 7×10-4 | 1.989 5 | |
| p2 | 20 | 7.591 4×10-4 | 4.373 1×10-4 | 2.204 2×10-4 | |||
| 40 | 9.387 9×10-5 | 3.015 5 | 5.404 4×10-5 | 3.016 5 | 2.734 7×10-5 | 3.010 8 | |
| 80 | 1.156 0×10-5 | 3.021 7 | 6.666 6×10-6 | 3.019 1 | 3.367 5×10-6 | 3.021 6 | |
| 160 | 1.444 0×10-6 | 3.001 0 | 8.328 3×10-7 | 3.000 9 | 4.206 3×10-7 | 3.001 1 |
| pk | grid | ‖eu‖L1 | Order | ‖eu‖L2 | Order | ‖eu‖L∞ | Order |
| p1 | 20 | 5.048 0×10-2 | 2.272 2×10-2 | 1.114 2×10-2 | |||
| 40 | 1.406 9×10-2 | 1.843 2 | 6.296 4×10-3 | 1.851 5 | 3.182 4×10-3 | 1.807 8 | |
| 80 | 3.750 0×10-3 | 1.907 6 | 1.675 3×10-3 | 1.910 1 | 8.612 9×10-4 | 1.885 6 | |
| 160 | 9.699 0×10-4 | 1.950 9 | 4.329 7×10-4 | 1.952 1 | 2.243 2×10-4 | 1.940 9 | |
| p2 | 20 | 8.871 8×10-4 | 4.526 9×10-4 | 2.763 0×10-4 | |||
| 40 | 1.002 9×10-4 | 3.145 1 | 5.405 3×10-5 | 3.066 1 | 3.017 3×10-5 | 3.194 9 | |
| 80 | 1.207 7×10-5 | 3.053 9 | 6.722 1×10-6 | 3.007 4 | 3.580 9×10-6 | 3.074 8 | |
| 160 | 1.475 8×10-6 | 3.032 6 | 8.358 5×10-7 | 3.007 6 | 4.338 9×10-7 | 3.045 0 |
Table 2 L1, L2 and L∞ errors for real part of u at time 1.0 in [0, 2π] with α = 0.5, β = γ = 1.0, A = c = 1 in Example 1 by using linear and quadratic elements
| pk | grid | ‖eu‖L1 | Order | ‖eu‖L2 | Order | ‖eu‖L∞ | Order |
| p1 | 20 | 5.048 0×10-2 | 2.272 2×10-2 | 1.114 2×10-2 | |||
| 40 | 1.406 9×10-2 | 1.843 2 | 6.296 4×10-3 | 1.851 5 | 3.182 4×10-3 | 1.807 8 | |
| 80 | 3.750 0×10-3 | 1.907 6 | 1.675 3×10-3 | 1.910 1 | 8.612 9×10-4 | 1.885 6 | |
| 160 | 9.699 0×10-4 | 1.950 9 | 4.329 7×10-4 | 1.952 1 | 2.243 2×10-4 | 1.940 9 | |
| p2 | 20 | 8.871 8×10-4 | 4.526 9×10-4 | 2.763 0×10-4 | |||
| 40 | 1.002 9×10-4 | 3.145 1 | 5.405 3×10-5 | 3.066 1 | 3.017 3×10-5 | 3.194 9 | |
| 80 | 1.207 7×10-5 | 3.053 9 | 6.722 1×10-6 | 3.007 4 | 3.580 9×10-6 | 3.074 8 | |
| 160 | 1.475 8×10-6 | 3.032 6 | 8.358 5×10-7 | 3.007 6 | 4.338 9×10-7 | 3.045 0 |
| pk | grid | ‖eu‖L2-LDPG | ‖eu‖L2-LDG | ‖eu‖L∞-LDPG | ‖eu‖L∞-LDG |
| p1 | 20 | 2.84×10-2 | 6.26×10-3 | 1.41×10-2 | 2.36×10-2 |
| 40 | 7.12×10-3 | 1.59×10-3 | 3.63×10-3 | 6.21×10-3 | |
| 80 | 1.79×10-3 | 3.90×10-4 | 9.22×10-4 | 1.49×10-3 | |
| 160 | 4.47×10-4 | 9.87×10-5 | 2.32×10-4 | 3.73×10-4 | |
| p2 | 20 | 4.37×10-4 | 1.24×10-4 | 2.20×10-4 | 6.56×10-4 |
| 40 | 5.40×10-5 | 2.07×10-5 | 2.73×10-5 | 2.45×10-4 | |
| 80 | 6.67×10-6 | 2.17×10-6 | 3.37×10-6 | 2.62×10-5 | |
| 160 | 8.33×10-7 | 2.52×10-7 | 4.21×10-7 | 3.09×10-6 |
Table 3 Comparison between LDPG method and LDG method: L2 and L∞ errors for real part of u at time 1.0 in [0, 2π] with α = 0.5, β = γ = 0, A = c = 1 in Example 1 by using linear and quadratic elements
| pk | grid | ‖eu‖L2-LDPG | ‖eu‖L2-LDG | ‖eu‖L∞-LDPG | ‖eu‖L∞-LDG |
| p1 | 20 | 2.84×10-2 | 6.26×10-3 | 1.41×10-2 | 2.36×10-2 |
| 40 | 7.12×10-3 | 1.59×10-3 | 3.63×10-3 | 6.21×10-3 | |
| 80 | 1.79×10-3 | 3.90×10-4 | 9.22×10-4 | 1.49×10-3 | |
| 160 | 4.47×10-4 | 9.87×10-5 | 2.32×10-4 | 3.73×10-4 | |
| p2 | 20 | 4.37×10-4 | 1.24×10-4 | 2.20×10-4 | 6.56×10-4 |
| 40 | 5.40×10-5 | 2.07×10-5 | 2.73×10-5 | 2.45×10-4 | |
| 80 | 6.67×10-6 | 2.17×10-6 | 3.37×10-6 | 2.62×10-5 | |
| 160 | 8.33×10-7 | 2.52×10-7 | 4.21×10-7 | 3.09×10-6 |
| pk | grid | ‖eu‖L2-LDPG | ‖eu‖L2-LDG | ‖eu‖L∞-LDPG | ‖eu‖L∞-LDG |
| p1 | 20 | 2.27×10-2 | 1.61×10-2 | 1.11×10-2 | 3.28×10-2 |
| 40 | 6.30×10-3 | 3.70×10-3 | 3.18×10-3 | 8.69×10-3 | |
| 80 | 1.68×10-3 | 9.58×10-4 | 8.61×10-4 | 1.93×10-3 | |
| 160 | 4.33×10-4 | 2.48×10-4 | 2.24×10-4 | 5.01×10-4 | |
| p2 | 20 | 4.53×10-4 | 1.48×10-4 | 2.76×10-4 | 7.90×10-4 |
| 40 | 5.41×10-5 | 1.88×10-5 | 3.02×10-5 | 1.04×10-4 | |
| 80 | 6.72×10-6 | 2.12×10-6 | 3.58×10-6 | 1.17×10-5 | |
| 160 | 8.36×10-7 | 2.42×10-7 | 4.34×10-7 | 1.32×10-6 |
Table 4 Comparison between LDPG method and LDG method: L2 and L∞ errors for real part of u at time 1.0 in [0, 2π] with α = 0.5, β = γ = 1.0, A = c = 1 in Example 1 by using linear and quadratic elements
| pk | grid | ‖eu‖L2-LDPG | ‖eu‖L2-LDG | ‖eu‖L∞-LDPG | ‖eu‖L∞-LDG |
| p1 | 20 | 2.27×10-2 | 1.61×10-2 | 1.11×10-2 | 3.28×10-2 |
| 40 | 6.30×10-3 | 3.70×10-3 | 3.18×10-3 | 8.69×10-3 | |
| 80 | 1.68×10-3 | 9.58×10-4 | 8.61×10-4 | 1.93×10-3 | |
| 160 | 4.33×10-4 | 2.48×10-4 | 2.24×10-4 | 5.01×10-4 | |
| p2 | 20 | 4.53×10-4 | 1.48×10-4 | 2.76×10-4 | 7.90×10-4 |
| 40 | 5.41×10-5 | 1.88×10-5 | 3.02×10-5 | 1.04×10-4 | |
| 80 | 6.72×10-6 | 2.12×10-6 | 3.58×10-6 | 1.17×10-5 | |
| 160 | 8.36×10-7 | 2.42×10-7 | 4.34×10-7 | 1.32×10-6 |
| grid | ‖eu‖L1 | Order | ‖eu‖L2 | Order | ‖eu‖L∞ | Order |
| 40 | 2.504 8×100 | 1.033 0×100 | 5.555 2×10-1 | |||
| 80 | 9.913 2×10-1 | 1.337 3 | 4.926 5×10-1 | 1.068 3 | 3.688 7×10-1 | 0.590 7 |
| 160 | 2.586 7×10-1 | 1.938 2 | 1.243 5×10-1 | 1.986 1 | 1.051 7×10-1 | 1.810 4 |
| 320 | 6.452 5×10-2 | 2.003 2 | 3.082 0×10-2 | 2.012 5 | 2.724 0×10-2 | 1.948 9 |
Table 5 L1, L2 and L∞ errors for real part of u at time 1.0 in [-15, 15] with LDPG1 in Example 2
| grid | ‖eu‖L1 | Order | ‖eu‖L2 | Order | ‖eu‖L∞ | Order |
| 40 | 2.504 8×100 | 1.033 0×100 | 5.555 2×10-1 | |||
| 80 | 9.913 2×10-1 | 1.337 3 | 4.926 5×10-1 | 1.068 3 | 3.688 7×10-1 | 0.590 7 |
| 160 | 2.586 7×10-1 | 1.938 2 | 1.243 5×10-1 | 1.986 1 | 1.051 7×10-1 | 1.810 4 |
| 320 | 6.452 5×10-2 | 2.003 2 | 3.082 0×10-2 | 2.012 5 | 2.724 0×10-2 | 1.948 9 |
| grid | ‖eu‖L1 | Order | ‖eu‖L2 | Order | ‖eu‖L∞ | Order |
| 40 | 8.479 6×10-1 | 3.678 9×10-1 | 1.912 1×10-1 | |||
| 80 | 5.060 9×10-2 | 4.066 5 | 2.097 5×10-2 | 4.132 6 | 2.319 7×10-2 | 3.048 6 |
| 160 | 3.811 7×10-3 | 3.730 9 | 1.535 8×10-3 | 3.771 6 | 1.720 0×10-3 | 3.748 2 |
| 320 | 3.577 9×10-4 | 3.413 3 | 1.394 2×10-4 | 3.461 5 | 1.446 8×10-4 | 3.571 2 |
Table 6 Numerical solutions at different μ with quadratic element in Example 2
| grid | ‖eu‖L1 | Order | ‖eu‖L2 | Order | ‖eu‖L∞ | Order |
| 40 | 8.479 6×10-1 | 3.678 9×10-1 | 1.912 1×10-1 | |||
| 80 | 5.060 9×10-2 | 4.066 5 | 2.097 5×10-2 | 4.132 6 | 2.319 7×10-2 | 3.048 6 |
| 160 | 3.811 7×10-3 | 3.730 9 | 1.535 8×10-3 | 3.771 6 | 1.720 0×10-3 | 3.748 2 |
| 320 | 3.577 9×10-4 | 3.413 3 | 1.394 2×10-4 | 3.461 5 | 1.446 8×10-4 | 3.571 2 |
| Time | 0 | 5 | 10 | 20 | 30 | 40 | 50 |
| Mass | 2.000 0 | 2.000 0 | 2.000 0 | 2.000 0 | 2.000 0 | 2.000 0 | 2.000 0 |
Table 7 Numerical mass in different time by using LDPG2 in Example 2
| Time | 0 | 5 | 10 | 20 | 30 | 40 | 50 |
| Mass | 2.000 0 | 2.000 0 | 2.000 0 | 2.000 0 | 2.000 0 | 2.000 0 | 2.000 0 |
Fig.1 Soliton propagation in Example 3 with initial condition (33) in [-25, 25], x0 = -10 and quadratic element with 200 uniform grids (single soliton case) (a) t = 0.0, (b) t = 2.0, (c) t = 5.0, (d) soliton evolution
Fig.2 Soliton collision in Example 3 with initial condition (34) in [-25, 25], x1 = -10, c2 = -4, x1 = 10 and quadratic element with 250 uniform grids (double soliton case) (a) t = 0.0, (b) t = 2.0, (c) t = 2.5, (d) t = 5.0
Fig.3 Simulation of the birth of standing and mobile soliton with initial conditons (35) and (36), A = 1.78 and in [-45, 45] and 400 uniform cells by using LDPG2 (a) standing and mobile, t = 0.0, (b) standing, t = 4.0, (c) mobile, t = 2.0, (d) mobile, t = 4.0
Fig.4 Bound state solution with N = 3 and initial condition (38) in Example 5 in [-15, 15] by LDPG2 with 500 uniform grids
Fig.5 Bound state solution with N = 4 and initial condition (38) in Example 5 in [-15, 15] by LDPG2 with 800 uniform grids
Fig.6 Bound state solution with N = 5 and initial condition (38) in Example 5 in [-15, 15] by LDPG2 with 1 000 uniform grids
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