JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (8): 67-76.doi: 10.6040/j.issn.1671-9352.0.2023.341

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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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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

CLC Number: 

  • O241

Table 1

The maximum absolute error calculated by two types of collocation method for different types of nodes (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

Table 2

The maximum absolute error and relative error calculated by two types of collocation method in different space-time domains (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

Table 3

The maximum absolute error calculated by two types of collocation method in different numbers of nodes (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

Fig.1

Numerical solution calculated by barycentric interpolation collocation method and solitary wave solution for equation"

Fig.2

The maximum error distribution of two types of collocation"

Table 4

The maximum absolute error calculated bybarycentric lagrange interpolation collocation method with different parameters a and p (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

Fig.3

The behavior of solitary wave solutions for equation with different parameters a, p"

Table 5

The maximum absolute error calculated by thebarycentric lagrange interpolation collocation method (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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