Traveling wave solutions of the nonlinear Gilson-Pickering equation in crystal lattice theory

doi:10.1016/j.joes.2022.06.009

Abstract

Abstract:

This paper focuses on obtaining the traveling wave solutions of the nonlinear Gilson-Pickering equation (GPE), which describes the prorogation of waves in crystal lattice theory and plasma physics. The solution of the GPE is approximated via the finite difference technique and the localized meshless radial basis function generated finite difference. The association of the technique results in a meshless approach that does not require linearizing the nonlinear terms. At the first step, the PDE is converted to a system of nonlinear ODEs with the help of the radial kernels. In the second step, a high-order ODE solver is adopted to discretize the nonlinear ODE system. The global collocation techniques pose a considerable computationl burden due to the calculation of the dense algebraic system. The proposed method approximates differential operators over the local support domain, leading to sparse differentiation matrices and decreasing the computational burden. Numerical results and comparisons are provided to confirm the efficiency and accuracy of the method.

Key words: Nonlinear Gilson-Pickering equation, Soliton wave solutions, Meshless technique, RBF, LRBF-FD, Optimal shape parameter

A.T. Nguyen, O. Nikan, Z. Avazzadeh. Traveling wave solutions of the nonlinear Gilson-Pickering equation in crystal lattice theory[J]. Journal of Ocean Engineering and Science, 2024, 9(1): 40-49.

A.T. Nguyen, O. Nikan, Z. Avazzadeh. [J]. Journal of Ocean Engineering and Science, 2024, 9(1): 40-49.

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URL: https://www.qk.sjtu.edu.cn/joes/EN/10.1016/j.joes.2022.06.009

https://www.qk.sjtu.edu.cn/joes/EN/Y2024/V9/I1/40

Figures/Tables 16

Algorithm 1. Full discretization of the GPE.

Algorithm 2. Optimal shape parameter [45].

Table 1.

L_{\infty}

norm errors, CN and CPUTime of Example 1 for Case I with

δ t = 0.001

when

T = 1

at various stencil sizes

n_{s}

$N$	$n_{s}$	$L_{\infty}$	CN	CPUTime
400	113	$9.5404 e - 05$	$4.2971 e + 05$	2.341001
400	119	$7.9713 e - 05$	$4.1670 e + 05$	$2.380281$
400	139	$2.3485 e - 05$	$3.5491 e + 05$	$2.420148$
400	155	$1.4817 e - 05$	$5.8869 e + 05$	$2.556396$
400	175	$1.4021 e - 05$	$4.9684 e + 05$	$2.603521$

Table 1.

L_{\infty}

norm errors, CN and CPUTime of Example 1 for Case I with

δ t = 0.001

when

T = 1

at various stencil sizes

n_{s}

$N$	$n_{s}$	$L_{\infty}$	CN	CPUTime
400	113	$9.5404 e - 05$	$4.2971 e + 05$	2.341001
400	119	$7.9713 e - 05$	$4.1670 e + 05$	$2.380281$
400	139	$2.3485 e - 05$	$3.5491 e + 05$	$2.420148$
400	155	$1.4817 e - 05$	$5.8869 e + 05$	$2.556396$
400	175	$1.4021 e - 05$	$4.9684 e + 05$	$2.603521$

Table 2.

L_{\infty}

L_{2}

, and

L_{rms}

norm errors and CPUTime when

δ t = 0.001

N = 400

and

n_{s} = 135

at different final times

T

in Case I over the spatial domain

[- 8, 8]

for Example 1.

$T$	Method [19]			LRBF-FD
	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{\infty}$	$L_{2}$	$L_{rms}$	CPUTime
0.5	$1.8425 e - 06$	$2.0140 e - 05$	$5.0304 e - 07$	$7.4506 e - 07$	$3.5181 e - 06$	$2.0312 e - 07$	0.325940
1.0	$7.2685 e - 06$	$9.3918 e - 05$	$2.3458 e - 06$	$1.3576 e - 06$	$6.7522 e - 06$	$3.8984 e - 07$	0.423046
1.5	$2.2759 e - 05$	$2.8776 e - 04$	$7.1872 e - 06$	$1.8230 e - 06$	$1.0367 e - 05$	$5.9856 e - 07$	0.524313
2.0	$6.9942 e - 05$	$8.9111 e - 04$	$2.2257 e - 06$	$3.4505 e - 06$	$1.6473 e - 05$	$9.5108 e - 07$	0.631450

Table 2.

L_{\infty}

L_{2}

, and

L_{rms}

norm errors and CPUTime when

δ t = 0.001

N = 400

and

n_{s} = 135

at different final times

T

in Case I over the spatial domain

[- 8, 8]

for Example 1.

$T$	Method [19]			LRBF-FD
	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{\infty}$	$L_{2}$	$L_{rms}$	CPUTime
0.5	$1.8425 e - 06$	$2.0140 e - 05$	$5.0304 e - 07$	$7.4506 e - 07$	$3.5181 e - 06$	$2.0312 e - 07$	0.325940
1.0	$7.2685 e - 06$	$9.3918 e - 05$	$2.3458 e - 06$	$1.3576 e - 06$	$6.7522 e - 06$	$3.8984 e - 07$	0.423046
1.5	$2.2759 e - 05$	$2.8776 e - 04$	$7.1872 e - 06$	$1.8230 e - 06$	$1.0367 e - 05$	$5.9856 e - 07$	0.524313
2.0	$6.9942 e - 05$	$8.9111 e - 04$	$2.2257 e - 06$	$3.4505 e - 06$	$1.6473 e - 05$	$9.5108 e - 07$	0.631450

Table 3.

L_{\infty}

L_{2}

, and

L_{rms}

norm errors and CPUTime by considering

δ t = 0.001

N = 300

and

n_{s} = 125

at different final times

T

in Case II over spatial domain

[- 8, 8]

for Example 1.

$T$	Method [19]			LRBF-FD
	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{\infty}$	$L_{2}$	$L_{rms}$	CPUTime
1	$9.0137 e - 07$	$5.6183 e - 06$	$1.4033 e - 07$	$9.8143 e - 09$	$3.9591 e - 08$	$2.2858 e - 09$	0.295395
2	$3.6118 e - 06$	$3.4581 e - 05$	$8.6371 e - 07$	$2.9988 e - 08$	$1.4725 e - 07$	$8.5016 e - 09$	0.327970
3	$1.1994 e - 05$	$1.1297 e - 04$	$2.8216 e - 06$	$7.0717 e - 08$	$3.9377 e - 07$	$2.2735 e - 08$	0.357182
4	$3.2036 e - 05$	$2.9399 e - 04$	$7.3428 e - 06$	$5.5848 e - 08$	$9.6731 e - 07$	$1.5208 e - 07$	0.381461
5	$7.7520 e - 05$	$7.0524 e - 04$	$1.7614 e - 05$	$3.7515 e - 07$	$2.1518 e - 06$	$1.2424 e - 07$	0.406676

Table 3.

L_{\infty}

L_{2}

, and

L_{rms}

norm errors and CPUTime by considering

δ t = 0.001

N = 300

and

n_{s} = 125

at different final times

T

in Case II over spatial domain

[- 8, 8]

for Example 1.

$T$	Method [19]			LRBF-FD
	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{\infty}$	$L_{2}$	$L_{rms}$	CPUTime
1	$9.0137 e - 07$	$5.6183 e - 06$	$1.4033 e - 07$	$9.8143 e - 09$	$3.9591 e - 08$	$2.2858 e - 09$	0.295395
2	$3.6118 e - 06$	$3.4581 e - 05$	$8.6371 e - 07$	$2.9988 e - 08$	$1.4725 e - 07$	$8.5016 e - 09$	0.327970
3	$1.1994 e - 05$	$1.1297 e - 04$	$2.8216 e - 06$	$7.0717 e - 08$	$3.9377 e - 07$	$2.2735 e - 08$	0.357182
4	$3.2036 e - 05$	$2.9399 e - 04$	$7.3428 e - 06$	$5.5848 e - 08$	$9.6731 e - 07$	$1.5208 e - 07$	0.381461
5	$7.7520 e - 05$	$7.0524 e - 04$	$1.7614 e - 05$	$3.7515 e - 07$	$2.1518 e - 06$	$1.2424 e - 07$	0.406676

Fig. 2. Motion of the single solitary wave with

N = 400, n_{s} = 145

and

δ t = 0.001

at different final times

T \in {0, 1, 2, 3, 4, 5, 6}

in Case I over spatial domain

[- 8, 8]

for Example 1.

Fig. 2. Motion of the single solitary wave with $N = 400, n_{s} = 145$ and $δ t = 0.001$ at different final times $T \in {0, 1, 2, 3, 4, 5, 6}$ in Case I over spatial domain $[- 8, 8]$ for Example 1.

Fig. 3. Motion of the single solitary wave with

N = 400, n_{s} = 145

and

δ t = 0.001

at various final times

T \in {0, 3, 4, 6, 8, 9, 12}

in Case II over spatial domain

[- 8, 8]

for Example 1.

Fig. 3. Motion of the single solitary wave with $N = 400, n_{s} = 145$ and $δ t = 0.001$ at various final times $T \in {0, 3, 4, 6, 8, 9, 12}$ in Case II over spatial domain $[- 8, 8]$ for Example 1.

Fig. 4. The soliton profiles of the GPE when

δ t = 0.001

N = 600

and

n_{s} = 251

in Case I over spatial domain

[- 8, 8]

for Example 1.

Fig. 4. The soliton profiles of the GPE when $δ t = 0.001$ , $N = 600$ and $n_{s} = 251$ in Case I over spatial domain $[- 8, 8]$ for Example 1.

Fig. 5. The soliton profiles of the GPE with

δ t = 0.001

N = 700

and

n_{s} = 451

in Case II over spatial domain

[- 8, 8]

for Example 1.

Fig. 5. The soliton profiles of the GPE with $δ t = 0.001$ , $N = 700$ and $n_{s} = 451$ in Case II over spatial domain $[- 8, 8]$ for Example 1.

Fig. 6. The contour of solutions and associated absolute errors when

δ t = 0.001

N = 500

and

n_{s} = 335

in Case I over spatial domain

[- 8, 8]

for Example 1.

Fig. 6. The contour of solutions and associated absolute errors when $δ t = 0.001$ , $N = 500$ and $n_{s} = 335$ in Case I over spatial domain $[- 8, 8]$ for Example 1.

Fig. 7. The contour of solutions and associated absolute errors with

δ t = 0.001

N = 400

and

n_{s} = 121

in Case II over the spatial domain

[- 8, 8]

for Example 1.

Fig. 7. The contour of solutions and associated absolute errors with $δ t = 0.001$ , $N = 400$ and $n_{s} = 121$ in Case II over the spatial domain $[- 8, 8]$ for Example 1.

Fig. 8. The sparsity patterns of the coefficient matrix

M

for two different stencil sizes

n_{s}

with

N = 100

Fig. 8. The sparsity patterns of the coefficient matrix $M$ for two different stencil sizes $n_{s}$ with $N = 100$ .

Table 4.

L_{\infty}

norm errors, CN and CPUTime of Example 1 with

δ t = 0.01

when

T = 10

for different stencil sizes.

$N$	$n_{s}$	$L_{\infty}$	CN	CPUTime
800	191	$6.8288 e - 08$	$1.7910 e + 06$	$0.520603$
800	219	$6.8284 e - 08$	$1.7910 e + 06$	$0.528558$
800	299	$6.7364 e - 08$	$2.3978 e + 06$	$0.529682$
800	609	$6.7167 e - 08$	$4.1260 e + 06$	$0.743456$
800	685	$6.7013 e - 08$	$1.6611 e + 06$	$0.697687$

Table 4.

L_{\infty}

norm errors, CN and CPUTime of Example 1 with

δ t = 0.01

when

T = 10

for different stencil sizes.

$N$	$n_{s}$	$L_{\infty}$	CN	CPUTime
800	191	$6.8288 e - 08$	$1.7910 e + 06$	$0.520603$
800	219	$6.8284 e - 08$	$1.7910 e + 06$	$0.528558$
800	299	$6.7364 e - 08$	$2.3978 e + 06$	$0.529682$
800	609	$6.7167 e - 08$	$4.1260 e + 06$	$0.743456$
800	685	$6.7013 e - 08$	$1.6611 e + 06$	$0.697687$

Table 5.

L_{\infty}

L_{2}

L_{rms}

, and

L_{re}

norm errors and CPUTime when

δ t = 0.001

N = 600

and

n_{s} = 425

at various final times

T

over the spatial domain

[- 8, 8]

for Example 2.

$T$	Method [19]				LRBF-FD
	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{re}$	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{re}$	CPUTime
1	$1.7273 e - 08$	$2.4865 e - 07$	$6.2104 e - 09$	$1.7274 e - 04$	$3.8678 e - 09$	$5.1417 e - 09$	$2.2994 e - 10$	$4.0174 e - 05$	0.431546
2	$3.3891 e - 08$	$4.8857 e - 07$	$1.2203 e - 08$	$3.3897 e - 04$	$7.6964 e - 09$	$1.0266 e - 08$	$4.5913 e - 10$	$8.0849 e - 05$	0.437426
3	$4.9899 e - 08$	$7.2176 e - 07$	$1.8027 e - 08$	$4.9914 e - 04$	$1.1587 e - 08$	$1.5530 e - 08$	$6.9454 e - 10$	$1.2169 e - 04$	0.444489
4	$6.4994 e - 08$	$9.4356 e - 07$	$2.3567 e - 08$	$1.5020 e - 03$	$1.5370 e - 08$	$2.0675 e - 08$	$9.2461 e - 10$	$1.6280 e - 04$	0.466018
5	$8.2218 e - 08$	$1.1909 e - 06$	$1.7614 e - 08$	$8.2260 e - 03$	$1.8929 e - 08$	$2.5521 e - 08$	$1.1413 e - 09$	$2.0419 e - 04$	0.467160

Table 5.

L_{\infty}

L_{2}

L_{rms}

, and

L_{re}

norm errors and CPUTime when

δ t = 0.001

N = 600

and

n_{s} = 425

at various final times

T

over the spatial domain

[- 8, 8]

for Example 2.

$T$	Method [19]				LRBF-FD
	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{re}$	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{re}$	CPUTime
1	$1.7273 e - 08$	$2.4865 e - 07$	$6.2104 e - 09$	$1.7274 e - 04$	$3.8678 e - 09$	$5.1417 e - 09$	$2.2994 e - 10$	$4.0174 e - 05$	0.431546
2	$3.3891 e - 08$	$4.8857 e - 07$	$1.2203 e - 08$	$3.3897 e - 04$	$7.6964 e - 09$	$1.0266 e - 08$	$4.5913 e - 10$	$8.0849 e - 05$	0.437426
3	$4.9899 e - 08$	$7.2176 e - 07$	$1.8027 e - 08$	$4.9914 e - 04$	$1.1587 e - 08$	$1.5530 e - 08$	$6.9454 e - 10$	$1.2169 e - 04$	0.444489
4	$6.4994 e - 08$	$9.4356 e - 07$	$2.3567 e - 08$	$1.5020 e - 03$	$1.5370 e - 08$	$2.0675 e - 08$	$9.2461 e - 10$	$1.6280 e - 04$	0.466018
5	$8.2218 e - 08$	$1.1909 e - 06$	$1.7614 e - 08$	$8.2260 e - 03$	$1.8929 e - 08$	$2.5521 e - 08$	$1.1413 e - 09$	$2.0419 e - 04$	0.467160

Fig. 9. The soliton profiles of the GPE when

δ t = 0.001

N = 650

and

n_{s} = 371

over the spatial domain

[- 8, 8]

in Example 2.

Fig. 9. The soliton profiles of the GPE when $δ t = 0.001$ , $N = 650$ and $n_{s} = 371$ over the spatial domain $[- 8, 8]$ in Example 2.

Fig. 10. The contour of solutions and associated absolute errors with

δ t = 0.001

N = 800

and

n_{s} = 401

over spatial domain

[- 8, 8]

in Example 2.

Fig. 10. The contour of solutions and associated absolute errors with $δ t = 0.001$ , $N = 800$ and $n_{s} = 401$ over spatial domain $[- 8, 8]$ in Example 2.

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