Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation

Jiaxiang Cai; Juan Chen; Min Chen

doi:10.3934/dcdsb.2021139

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2022, Volume 27, Issue 5: 2441-2453. Doi: 10.3934/dcdsb.2021139

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Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation

1.
School of Geography, Nanjing Normal University, Nanjing, Jiangsu, 210023, China
2.
School of Mathematics and Statistics, Huaiyin Normal University, Huaian, Jiangsu 223300, China
3.
Department of Basic Education, Jiangsu Vocational College of Finance & Economics, Huaian, Jiangsu, 223003, China

^* Corresponding author: cjx1981@hytc.edu.cn (J. Cai)
^* Corresponding author: cjx1981@hytc.edu.cn (J. Cai)

Received: November 2020

Revised: March 2021

Early access: May 2021

Published: May 2022

The first author is supported by the Natural Science Foundation of Jiangsu Province of China grant BK20181482, China Postdoctoral Science Foundation through grant 2020M671532, Jiangsu Province Postdoctoral Science Foundation through grant 2020Z147 and Qing Lan Project of Jiangsu Province of China

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Abstract

A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with the Kahan's method in time and symplectic Euler-box rule in space. It can be implemented easily, and also it is less storage-consuming and more efficient than the fully implicit methods. Several numerical experiments, including simulations of evolution of the line-soliton and lump-type soliton and interaction of the two lumps, are carried out to show the good performance of the scheme.

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Mathematics Subject Classification: 65P10, 65N35, 65N06.

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Figure 1. Left: the errors in solution (Solid line: $ {\rm e}_{\infty} $; Dashed line: $ {\rm e}_2 $); Right: waveform at time $ t = 12 $

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Figure 2. The lump type solitary wave at time $ t = 0 $ (top), $ t = 0.5 $ (the second row), $ t = 1 $ (the third row) and $ t = 5 $ (bottom), respectively. Left: the profiles of solution; Middle: contours of numerical solution; Right: contours of exact solution

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Figure 3. The lump type solitary wave at time $ t = 20 $. Left: the profile of solution; Middle: contours of numerical solution; Right: contours of exact solution

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Figure 4. The profiles of interaction of the two lump waves at different times and the variation of the global energy (the last graph): $ \Delta t = 1 $e-2, $ \Delta x = \Delta y = 2 $e-1

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Figure 5. The motion of single soliton for the KPII equation. Top: $ t = 0 $; Bottom: $ t = 8 $

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Table 1. Numerical results for the KPI equation obtained by the present scheme: $ \Omega = [0,40]\times[0,2] $, $ r = (\Delta t,\Delta x,\Delta y) = (0.1,0.2,0.2) $ and $ t = 1 $

Mesh	$r$	$r/2$	$r/2^2$	$r/2^3$	$r/2^4$
${\rm e}_{\infty}$	4.4921e-2	1.1041e-2	2.8764e-3	7.1999e-4	1.6418e-4
Rate	-	2.0248	1.9405	1.9982	2.1327
${\rm e}_2$	5.8460e-2	1.3052e-2	3.2214e-3	8.0842e-4	1.9908e-4
Rate	-	2.1632	2.0185	1.9945	2.0218

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