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《山东大学学报(理学版)》 ›› 2024, Vol. 59 ›› Issue (8): 67-76.doi: 10.6040/j.issn.1671-9352.0.2023.341

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重心插值配点法求解小振幅长波广义BBM-KdV方程

吕秀敏1(),葛倩2,*(),李金2   

  1. 1. 山东交通学院理学院，山东 济南 250357
    2. 山东建筑大学理学院，山东 济南 250101
  • 收稿日期:2023-08-04 出版日期:2024-08-20 发布日期:2024-07-31
  • 通讯作者: 葛倩 E-mail:lvxiumin@sdjtu.edu.cn;geqian@sdjzu.edu.cn
  • 作者简介:吕秀敏(1986—)，女，讲师，硕士，研究方向为微分、积分方程数值解. E-mail：lvxiumin@sdjtu.edu.cn
  • 基金资助:
    山东省自然科学基金资助项目(ZR2022MA003);山东交通学院科研基金资助项目(Z202334)

Barycentric interpolation collocation method for solving the small-amplitude long-wave scheme generalized BBM-KdV equation

Xiumin LYU1(),Qian GE2,*(),Jin LI2   

  1. 1. School of Science, Shandong Jiaotong University, Jinan 250357, Shandong, China
    2. School of Science, Shandong Jianzhu University, Jinan 250101, Shandong, China
  • Received:2023-08-04 Online:2024-08-20 Published:2024-07-31
  • Contact: Qian GE E-mail:lvxiumin@sdjtu.edu.cn;geqian@sdjzu.edu.cn

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摘要：

应用重心插值配点法求解小振幅长波格式下的广义Benjamin-Bona-Mahony(BBM)-Korteweg-de Vries(KdV)方程。针对方程中的非线性项ηpηx，采用直接线性化方法将其转化为线性项。利用重心插值基函数构造方程未知函数的近似函数，建立时空域上重心插值配点法离散广义BBM-KdV方程的矩阵方程，并进行了收敛性分析。数值算例验证了重心插值配点法求解广义BBM-KdV方程的有效性和数值计算精度，其计算精度可达到10-8量级。

关键词: 小振幅长波, 广义BBM-KdV方程, 重心插值, 配点法

Abstract:

The barycentric interpolation collocation method is applied to solve the small-amplitude long-wave scheme generalized Benjamin-Bona-Mahony(BBM)-Korteweg-de Vries (KdV) equation in the small amplitude long wave scheme. By the direct linearization method, the nonlinear term ηpηx in the equation is converted into a linear term. Unknown functions of BBM-KdV equation is approximated by the barycentric interpolation basis function. The matrix equation of the generalized BBM-KdV equation is obtained by discretized the generalized BBM-KdV equation in the space-time domain, and the convergence analysis is carried out. The effectiveness and numerical accuracy of the barycentric interpolation collocation method is verified by numerical example, and the calculation accuracy can reach the order of 10-8.

Key words: small-amplitude long-wave, generalized BBM-KdV equation, barycentric interpolation, collocation method

中图分类号: 

  • O241

表1

两类配点法分别取不同类型节点计算的最大绝对误差(x ∈ [－10, 10], t ∈ [0, 10], α=00.05)"

m×n $ E_{\mathrm{r} \text { max }}^{\mathrm{c}}$ $ E_{\mathrm{r} \text { max }}^{\mathrm{d}}$ $ E_{\mathrm{l} \text { max }}^{\mathrm{c}}$ $ E_{\mathrm{l} \text { max }}^{\mathrm{d}}$ p
15×10 4.000 2×10-11 1.576 8×10-10 2.553 6×10-13 1.815 1×10-12
15×20 2.591 3×10-12 1.789 5×10-11 5.999 5×10-14 7.877 6×10-12 1
25×20 1.220 4×10-15 3.236 9×10-12 1.648 0×10-16 7.270 3×10-09
15×10 7.066 4×10-11 2.966 8×10-10 2.445 5×10-13 1.235 5×10-12
15×20 2.653 5×10-12 1.954 0×10-11 3.816 4×10-14 1.203 2×10-10 3
25×20 1.815 2×10-14 7.216 8×10-11 4.052 3×10-15 3.418 9×10-08

表2

两类配点法分别取不同时空域计算的最大绝对误差和相对误差(a=0.05, p=3, mn=45×30)"

空间域 时间域 Er maxc El maxc Rrerc Rlerc
[-10, 10] [0, 10] 4.053 2×10-12 1.458 6×10-14 4.249 2×10-12 2.169 1×10-14
[0, 15] 4.976 1×10-12 1.136 6×10-14 4.858 0×10-12 1.913 8×10-14
[0, 20] 4.405 3×10-12 9.756 1×10-15 3.231 7×10-12 1.747 2×10-14
[0, 25] 2.716 6×10-12 6.800 1×10-15 2.679 5×10-12 1.569 5×10-14
[0, 30] 2.655 5×10-12 6.966 6×10-15 2.426 5×10-12 1.108 4×10-14
[-20, 20][0, 10] 3.365 1×10-12 1.038 1×10-14 4.559 9×10-12 2.045 1×10-14
[0, 15] 2.632 0×10-12 9.131 6×10-15 2.431 4×10-12 1.879 2×10-14
[0, 20] 8.674 5×10-12 1.013 1×10-14 1.037 9×10-11 1.822 8×10-14
[0, 25] 3.724 0×10-12 9.992 0×10-15 5.183 6×10-12 2.275 2×10-14
[0, 30] 3.742 9×10-12 7.993 6×10-15 4.067 0×10-12 1.663 0×10-14
[-30, 30][0, 10] 1.959 0×10-12 1.755 5×10-14 2.696 9×10-12 2.863 9×10-14
[0, 15] 2.061 0×10-12 7.875 6×10-15 2.231 7×10-12 1.856 6×10-14
[0, 20] 4.882 6×10-12 6.855 6×10-15 6.085 3×10-12 1.787 9×10-14
[0, 25] 4.124 1×10-12 7.841 0×10-15 5.690 9×10-12 1.616 8×10-14
[0, 30] 2.750 3×10-12 1.020 0×10-14 3.798 8×10-12 2.205 6×10-14

表3

两类配点法分别取不同数量节点计算的最大绝对误差(a=0.05, x ∈ [－30, 30], t ∈ [0, 10])"

m×n p=1 p=2 p=3
Er maxc El maxc Er maxc El maxc Ermaxc El maxc
35×25 3.774 8×10-14 5.665 6×10-15 2.602 1×10-14 3.934 4×10-15 1.317 8×10-13 9.839 4×10-15
40×25 1.026 4×10-14 3.024 9×10-16 2.691 0×10-13 3.840 7×10-15 6.600 6×10-13 6.633 6×10-15
45×25 5.688 7×10-14 4.651 2×10-16 4.350 8×10-13 6.272 8×10-15 1.329 8×10-12 1.404 4×10-14
50×25 6.026 3×10-13 2.958 8×10-16 6.728 9×10-12 3.639 4×10-15 1.286 3×10-11 1.280 9×10-14
45×35 1.617 2×10-13 1.234 9×10-15 8.871 1×10-13 1.030 1×10-14 3.281 0×10-12 2.771 4×10-14
45×40 2.527 6×10-13 1.298 0×10-15 1.876 9×10-12 1.202 9×10-14 4.171 4×10-12 3.194 7×10-14
45×45 1.968 2×10-13 1.708 7×10-15 5.312 9×10-12 1.731 9×10-14 1.563 3×10-11 4.436 7×10-14
45×50 2.228 7×10-12 1.855 7×10-15 1.023 2×10-11 1.806 9×10-14 2.992 7×10-11 3.834 4×10-14

图1

重心插值配点法的数值解与方程的孤波解"

图2

两类配点法的最大误差分布"

表4

取不同参数a、p时重心Lagrange插值配点法计算的最大绝对误差(mn=45×30)"

空间域 时间域 a=0.05, p=1 a=0.1, p=1 a=0.1, p=3
[-10, 10][0, 10] 6.539 9×10-16 2.302 0×10-15 2.861 6×10-14
[0, 15] 4.566 7×10-16 1.486 7×10-15 1.695 9×10-14
[0, 20] 3.937 8×10-16 7.580 7×10-16 9.478 5×10-15
[0, 25] 4.352 0×10-16 2.086 5×10-14 1.280 9×10-14
[0, 30] 2.693 2×10-16 6.853 5×10-13 1.099 4×10-13
[-20, 20][0, 10] 5.299 6×10-16 1.749 6×10-14 1.018 6×10-14
[0, 15] 3.987 7×10-16 1.673 3×10-14 1.342 7×10-14
[0, 20] 3.664 6×10-16 1.651 8×10-14 1.960 9×10-14
[0, 25] 4.098 3×10-16 3.007 0×10-14 1.046 4×10-14
[0, 30] 3.996 4×10-16 1.922 0×10-12 2.898 2×10-13
[-30, 30][0, 10] 6.844 6×10-16 2.202 7×10-10 3.639 3×10-11
[0, 15] 3.688 5×10-16 2.084 2×10-10 3.552 2×10-11
[0, 20] 2.797 2×10-16 2.072 4×10-10 3.464 8×10-11
[0, 25] 2.634 1×10-16 1.986 5×10-10 3.266 3×10-11
[0, 30] 4.006 7×10-16 1.819 3×10-10 3.087 1×10-11

图3

不同a、p时方程孤波解的性态"

表5

重心Lagrange插值配点法计算的最大绝对误差(a=0.3, p=1, x ∈ [－10, 10])"

m×n t∈[0, 10] t∈[0, 15] t∈[0, 20] t∈[0, 25] t∈[0, 30]
45×35 2.505 7×10-9 2.564 0×10-9 2.337 9×10-7 3.182 9×10-6 1.348 3×10-5
45×45 2.693 5×10-9 2.318 6×10-9 2.457 7×10-9 5.674 0×10-8 4.768 5×10-7
45×50 2.778 7×10-9 2.338 7×10-9 2.219 0×10-9 7.469 1×10-9 8.791 2×10-8
50×45 2.237 4×10-10 1.889 0×10-10 2.219 1×10-9 5.983 1×10-8 4.868 6×10-7
50×50 2.287 3×10-10 1.919 8×10-10 1.942 5×10-10 7.411 5×10-9 9.292 5×10-8
50×55 2.262 3×10-10 1.951 1×10-10 1.850 5×10-10 1.325 6×10-9 1.673 7×10-8
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