| Based on the basic rule that numerical algorithms should preserve the intrin-sic properties of the original problems as much as possible, Feng first presented the concept of symplectic schemes for Hamiltonian systems and further the structure-preserving algorithms for the general conservative dynamical systems. Nowadays,structure-preserving algorithms have gained remarkable success in the numerical anal-ysis for finite dimensional Hamiltonian systems because of their excellent stability and accurate long-time numerical behavior. However,numerous practical problems such as the propagation of nonlinear waves and the evolution of electromagnetic field-s often involve infinite dimensional Hamiltonian systems. The structure-preserving algorithms for such systems are not perfect yet and still in the initial stage, where there are many issues about fundamental theories and practical applications waiting to be solved. Therefore, we devote this thesis further to investigating the structure-preserving algorithms for infinite dimensional Hamiltonian systems mainly including the following two parts. In the first part, we will develop the basic theory of structure-preserving algorithms for infinite dimensional Hamiltonian systems continuously. On the one hand, there exist a few studies on the local structure-preserving algorithm af-ter it was put forward. First, we take the Korteweg-de Vries equation as an example to propose a series of local structure-preserving algorithms. Then,we systematically give a unified framework to construct a series of local structure-preserving algorithm-s for the general conservative systems both in one- and two-dimension, which can be applied for a huge class of partial differential equations (PDEs). On the other hand, as there is not too much work about the traditional error estimate of structure-preserving algorithms, based on the different spatial discretization methods, we con-struct and analyze some new structure-preserving algorithms for the coupled non-linear Schrodinger equation and Klein-Gordon-Schrodinger equation with different boundary conditions. The second part is for the applications of the new constructed algorithms to the simulations of nonlinear waves. The main results are as follows.1. Because the structures of PDEs are defined on the global time level, they will inevitably depend on the boundary conditions. Therefore, the necessary conditions for applying the traditional symplectic, energy-preserving or momentum-preserving algorithms to the given PDEs are not only the conservative system itself but also the proper boundary conditions. In order to enlarge the range of application of the structure-preserving algorithms, based on the concatenating method, we present a u-nified framework to construct a series of local structure-preserving algorithms for the Korteweg-de Vries equation, including eight multi-symplectic algorithms, eight lo-cal energy-preserving algorithms and eight local momentum-preserving algorithms.Among these algorithms,some have been discussed and widely used before while the most are novel schemes. The outstanding advantage of these proposed algo-rithms is that they conserve the corresponding discrete local structures in any time-space region exactly. With proper boundary conditions, such as periodic or homoge-neous boundary conditions, the local structure-preserving algorithms will be global structure-preserving algorithms. Linear stability and numerical experiments are con-ducted to show the performance of the proposed methods. The unified framework can be easily applied to many other equations.2. Many PDEs can be written as a multi-symplectic Hamiltonian system, which has three local conservation laws, namely multi-symplectic conservation law, local en-ergy conservation law and local momentum conservation law. Although some studies indicate that the local structure-preserving algorithm can be applied to some special e-quations, there still exist many other conservative PDEs need to be verified. Therefore,we systematically give a unified framework to construct the local structure-preserving algorithms for the general conservative PDEs both in one-and two-dimension starting from the multi-symplectic formulation and using the concatenating method. These proposed local structure-preserving algorithms include the famous multi-symplectic Preissmann scheme, Euler-box scheme and many new schemes. Besides, we not only use the average vector field (AVF) method to construct local energy-preserving algo-rithms for spatial discretization but also use it to construct local momentum-preserving algorithms for temporal discretization. It is worth noting that the local structure-preserving algorithms proposed here are all independent of the boundary conditions and can be applied for a huge class of conservative PDEs. These algorithms are il-lustrated by the nonlinear Schrodinger equation and the Klein-Gordon-Schrodinger equation. Numerical experiments are conducted to show the good performance of the proposed methods.3. Due to the convenience of implementation on machines, finite difference methods are popular for the Hamiltonian systems. Recently, the high-order differ-ence methods play an important role in the simulation of high frequency wave phe-nomena. we propose and analyze a two-level conservative high-order compact finite difference scheme for the Klein-Gordon-Schrodinger equation with Dirichlet bound-ary condition. We use the time-derivative of the real scalar meson solution?(x, t)as a separate dependent variable and reformulate the initial-boundary value problem into an equivalent system, which overcomes the difficulty of analyzing the two-level compact scheme for this equation. What's more, we convert the point-wise form of the new scheme into an equivalent vector form to solve the problem caused by the compact operator and the Dirichlet boundary condition. Then, we apply the energy method and matrix knowledge on the vector form to prove that the new scheme pre-serves the total charge and energy in the discrete level and obtain the convergence rate of the new scheme, without any restrictions on the grid ratio, at the order of (?)(h4??2)in L2 -norm. Numerical experiments are carried out to verify the performance of the scheme.4. Compared to the large amount of studies in finite difference methods, Fouri-er pseudospectral method also has been one of the dominant players on platforms of method of lines in past two decades due to its high accuracy and efficiency, especially in the simulation of high frequency wave phenomena. We derive an efficient con-servative scheme for the coupled nonlinear Schrodinger system, based on the Fourier pseudospectral method, the Crank-Nicolson method and leap-frog method. Our key idea consists of two aspects. First, we solve the coupled nonlinear Schrodinger sys-tem based on its Hamiltonian structure and the resulted scheme can still preserve the Hamiltonian nature. Second, we use Fourier pseudospectral method in spatial dis-cretization and Crank-Nicolson/ leap-frog scheme for discretizing linear/ nonlinear terms in time direction, respectively. The proposed scheme is then energy-preserving,mass-preserving, uniquely solvable and unconditionally stable, while being decou-pled, linearized and suitable for parallel computation in practical implementation. Us-ing the energy method and the classical interpolation theory, an error estimate of the proposed scheme is proven strictly without any grid ratio restrictions in the discrete L2-norm. Finally, numerical results are reported to support our theoretical analysis.5. We also focus on constructing a new Fourier pseudospectral conservative scheme for Klein-Gordon-Schrodinger equation and analyzing its conservative and convergent properties in the dissertation. We first rewrite the equation as an infinite-dimensional Hamiltonian system. Secondly, we use the idea of method of lines to obtain a semi-discretization induced by the Fourier pseudospectral method,which can be written as a canonical finite-dimensional Hamiltonian form. Then, the energy-preserving and charge-preserving scheme is constructed by using the symmetric dis-crete gradient method. Thirdly,based on the discrete conservation law and the e-quivalence between the semi-norm in the Fourier pseudospectral method and that in the finite difference method,the pseudospectral solution of the proposed scheme is proved to be bounded in the discrete L?-norm. Then, the proposed scheme is shown to be convergent with the convergence order of (?)(J-r + ?2) in the discrete L2-norm afterwards. Numerical results are in agreement with the theoretical analysis. |
PDF Full Text Request