Multi-symplectic methods for membrane free vibration equation

Abstract

In this paper, the multi-symplectic formulations of the membrane free vibration equation with periodic boundary conditions in Hamilton space are considered. The complex method is introduced and a semi-implicit twenty-seven-points scheme with certain discrete conservation laws a multi-symplectic conservation law (CLS), a local energy conservation law (ECL) as well as a local momentum conservation law (MCL) is constructed to discrete the PDEs that are derived from the membrane free vibration equation. The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior.

Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation

“…For the Landau-Ginzburg-Higgs equation, we can simplify the corresponding expression and obtain the local momentum conservation law [3][4][5][6] ,…”

“…Based on the local properties of some Hamilton systems, many conservative PDEs, such as the Schrödinger equations and the KdV equations, which allow the multi-symplectic formulations with the exact conservation laws, are extensively investigated in Refs. [1][2][3][4][5][6]. In this paper, the multi-symplectic Runge-Kutta method for the Landau-GinzburgHiggs equation is studied in detail.…”

The complex multi-symplectic scheme for the generalized sinh-Gordon equation

“…The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior and high accuracy. generalized sinh-Gordon equation, multi-symplectic, complex method, Runge-Kutta methodsThe multi-symplectic integrator, proven to be a very robust framework for the accurate, efficient and longtime integration of some nonlinear evolution equations, was widely investigated during the last decade [1][2][3][4][5][6][7][8][9][10][11][12] . Bridges, Reich and Moore presented the concept of the multi-symplectic integrator and applied it to solving nonlinear wave equation [1][2][3][4] and nonlinear Schrödinger equation [2] .…”

“…Bridges, Reich and Moore presented the concept of the multi-symplectic integrator and applied it to solving nonlinear wave equation [1][2][3][4] and nonlinear Schrödinger equation [2] . Subsequently, the multi-symplectic schemes were constructed to obtain the solutions to several physically important nonlinear evolution equations, such as membrane free vibration equation [5] , Boussinesq equation [7] , KdV equation [8][9][10] , Schrödinger equation [11] , Ginzburg-Landau equation [12] and so on, numerically.The generalized sinh-Gordon equation, first appearing in the propagation of fluxons in Josephson junctions between two superconductors [13] , then in such fields as differential geometry, solid state physics, nonlinear optics, and dislocations in metals, has been theoretically studied to a great extent [14,15] . However, the numerical method for sinh-Gordon equation has not been reported formally.…”

“…Bridges, Reich and Moore presented the concept of the multi-symplectic integrator and applied it to solving nonlinear wave equation [1][2][3][4] and nonlinear Schrödinger equation [2] . Subsequently, the multi-symplectic schemes were constructed to obtain the solutions to several physically important nonlinear evolution equations, such as membrane free vibration equation [5] , Boussinesq equation [7] , KdV equation [8][9][10] , Schrödinger equation [11] , Ginzburg-Landau equation [12] and so on, numerically.…”