Several Kinds of Soliton Solution of Nonlinear Schrödinger Equation: Local Discontinuous Petrov-Galerkin Method

doi:10.19596/j.cnki.1001-246x.8515

Abstract

Abstract:

A local discontinuous Petrov-Galerkin method is developed for nonlinear Schrödingerequations. Several kinds of solitons are simulated and related phenomena are discussed, such as the soliton propagation and collision, birth of solitons including standing soliton and mobile soliton, the bound state of N solitons. The algorithm simulates some narrow structures in soliton related phenomenon. Numerical examples show that the algorithm has high accuracy and can reach the optimal convergence order. Compared with local discontinuous Galerkin method, the local discontinuous Petrov-Galerkin method has high computational efficiency and simple computational formula.

Key words: local discontinuous Petrov-Galerkin method, nonlinear Schrödinger equation, soliton, bound state of N solitons

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.

Figures/Tables 13

References 21

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p^k	grid	‖e_u‖_L¹	Order	‖e_u‖_L²	Order	‖e_u‖_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

p^k	grid	‖e_u‖_L¹	Order	‖e_u‖_L²	Order	‖e_u‖_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

p^k	grid	‖e_u‖_L¹	Order	‖e_u‖_L²	Order	‖e_u‖_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

p^k	grid	‖e_u‖_L¹	Order	‖e_u‖_L²	Order	‖e_u‖_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

p^k	grid	‖e_u‖_L²-LDPG	‖e_u‖_L²-LDG	‖e_u‖_L^∞-LDPG	‖e_u‖_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