# American Institute of Mathematical Sciences

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)

Received  November 2020 Revised  March 2021 Early access May 2021

Fund Project: 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

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.

Citation: Jiaxiang Cai, Juan Chen, Min Chen. Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021139
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##### References:
Left: the errors in solution (Solid line: ${\rm e}_{\infty}$; Dashed line: ${\rm e}_2$); Right: waveform at time $t = 12$
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
The lump type solitary wave at time $t = 20$. Left: the profile of solution; Middle: contours of numerical solution; Right: contours of exact solution
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
The motion of single soliton for the KPII equation. Top: $t = 0$; Bottom: $t = 8$
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
 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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