Symplectic And Multi-symplectic Algorithms To Some Nonlinear Evolution Equations

We consider symplectic and multisymplectic algorithms for some evolution equations in the dissertation. We study the evolution phenomena of solitary waves for the symmetry regular long wave equation (SRLW) and the Klein-Gordon-Schrodinger (KGS) equations by symplectic and multisymplectic integrators. Moreover, the associated conservation laws are discussed.The solitary waves and their theory are important part of modern nonlinear scientific research; and their mathematical models are sometimes linear or nonlinear partial differential equations. They arise in many practical fields, such as: fluid dynamics, physics of atom and molecule, plasma physics, optical fiber communication, chemistry, biology and medicine. Finding the exact solitary wave solutions for the physical problems and their theory have become hot questions for discussion in the last decades. Plentiful and substantial achievements have been made, which motivate a lot of methods to obtain the exact solutions of solitary wave equations, such as: the homogeneous balance method, the hyperbolic function method, the power series method, besides the traditional methods, such as: the inverse scatting method, the bilinear Hirota method, and the Backlund transformation method etc. However, most of the solutions for the determined problems can not be exactly solved for the complexity of things. And we can only obtain their approximate solutions by numerical methods, especially for the nonlinear cases.All the physical courses whose dissipative effects are negligible can be expressed as Hamiltonian systems which preserve symplectic geometric structure. The Hamiltonian system is universal in the nature, in other words, most of soliton equations can be written into Hamiltonian formalism. The basic principle of modern numerical computation is to preserve the intrinsical character of the original problems. Therefore, it is necessary to study numerical methods which preserve the symplectic structure of the Hamiltonian system. Academician Kang Feng, the fou- nder of Chinese computational mathematics, put forward the symplectic geometric algorithm systemically which preserves symplectic structure of the Hamiltonian system in 1984. Symplectic integrator became a hot question for discussion in computational science consequently in domestic and foreign, and numbers of related achievements welled up. Since symplectic algorithms were brought forward by Kang Feng, two overflying developments took on: one is the generalization from finite dimensional Hamiltonian system to the infinite dimensional, the other is the generalization to multisymplectic algorithms. In the later 1990s, Marsden etal introduced the concept of multisymplectic integrator from variational principle, and Bridges & Reichs exported the multisymplectic integrator from the aspect of symplectic geometry. Rapid progress has been made in the recent years. It has solved many practical problems successfully and simulated lots of physical phenomena.In chapter 1, we introduce the background of symplectic and multisymplectic algorithms, and review the preliminary knowledge about the symplectic space.In chapter 2, the commonly used methods to construct symplectic algorithms are summarized, including the generating function method, the Runge-Kutta methods, the partition Runge-Kutta method, the explicit method for the separable Hamiltonian system, and the composition method to construct high order scheme, etc. The Fourier pseudo-spectral method to lower the infinite dimensional Hamiltonian system to the finite dimension is introduced to end the chapter.In chapter 3, a family of symplectic schemes are constructed for the KGS equations, and their conservation laws and convergence ratio are analyzed. We prove that the symplectic schemes preserve the charge conservation law exactly and analyze the energy residual. Both the global error and convergence ratio of the numerical solution are of (?)(Ï„² + h^2m). Numerical illustrations show that the symplectic schemes we construct are capable of simulating the original problems in a long time. Moreover, they demonstrate the correctness of theoretical analysis.In chapter 4, the multisymplectic Hamiltonian system and its associated conservation laws are simply presented. For simplicity, we take the KGS system as example to introduce the commonly used methods to es- tablish multisymplectic integrators. The main methods are the Fourier pseudo-spectral method and the Gauss-Legendre Runge-Kutta method etc. The conservation laws are studied for the KGS system. We discover that the Preissman scheme conserves the charge in the sense of weight, and the multisymplectic Fourier pseudo-spectral method preserves the charge in the classic sense. However, none of they can preserve the energy because the expression of energy is a 3 degree polynomial. Fortunately, the residual of energy is relative small. Numerical examples show that the schemes can simulate various solitary waves. A lot of interesting physical phenomena are observed. The correctness of our theoretical analysis is simultaneity illustrated.In chapter 5, the multisymplectic integrators for the SRLW equation are discussed, which include the multisymplectic Preissman scheme and the Fourier pseudo-spectral method. Numerical examples show the validity and superiority of the schemes we construct. We also study the methods to modify the multisymplectic scheme to a local energy conservation law scheme or a local momentum conservation law scheme, which take SRLW equation as example to illustrate the methods.In the last chapter, we summarize the main conclusions of the dissertation. The future prospects and some challenges as well as some open problems we face with symplectic and multisymplectic algorithms are listed in the end.