Structure-Preserving Algorithms For Oscillatory Hamiltonian Systems

This thesis is concerned with structure-preserving algorithms for oscillatory Hamil-tonian systems.It is well-known that Hamiltonian systems form a significant category of dynamical systems. Any real physical process with negligible dissipation can be described by a Hamiltonian system. This system has broad applications, which include the structural biology, pharmacology, semiconductivity, superconductivity, plasma physics, celestial mechanics, material mechanics, etc. The first five topics have been listed as " Grand Challenges" in Research Project of American government. One of the founders of the quantum mechanics, Schrodinger said," Hamiltonian principle has been the foundation for modern physics …. If you want to solve any physics problem using the modern the-ory, you must present it using the Hamiltonian formulation" [99]. Thus, it is important to investigate the structure-preserving numerical methods for Hamiltonian systems.Hamiltonian systems have two important properties:the energy conservation and symplectic structure. Traditional high-order integrators minimize the error brought by approximation, but in general they miss some important structures of the systems. As a result, powerful in simulation of physical phenomena as they are, these methods are usually limited to the scope of short term computation. Therefore, devising numerical methods that preserve the intrinsic physical properties (e.g. conservation of energy or momentum) or geometric properties (e.g. symplectic or multi-symplectic structure) in long term integration has turned out to be in urgent need. In recent years, symplectic schemes for Hamiltonian systems of ordinary differential equations (ODEs) are system-atically and extensively explored. Theoretical and practical research suggests that sym-plectic integrators behave much better than nonsymplectic integrators, especially over long time. Meanwhile, as a natural generalization of Hamiltonian ODEs, Hamiltonian partial differential equations (PDEs) are proposed with multi-symplectic structures in both temporal and spatial directions. One of the great challenges in the numerical solu-tion of PDEs is the development of robust and stable algorithms. Similar to symplectic methods for Hamiltonian ODEs, multi-symplectic integrators are supposed to preserve the multi-symplectic structures of Hamiltonian PDEs. The research on symplectic ge-ometry (symplectic and multi-symplectic) is extremely enriched and vital. Besides, as another important invariant of Hamiltonian system, energy-preserving is more impor-tant in some aspects. At present, more and more researchers are focusing on designing methods to preserve the original Hamiltonian energy.On the other hand, in pure and applied mathematics, physics, astronomy, chemistry, molecular biology, classical and quantum physics, mechanics, the following second-order oscillatory system is frequently encountered, where M is a d x d positive semidefinite matrix which im-plicitly contains the frequencies of the oscillatory problem (1) and f:R × Rd→ Rd is sufficiently smooth. This system has remarkable structure because of the presence of the linear term My. The solution of (2) is a nonlinear multi-frequency oscillator. Be-sides, if M is symmetric and positive semidefinite, and f(y(t))=-▽yU(y(t)), the corresponding system (2) is a Hamiltonian system. However, traditional numerical methods such as Runge-Kutta type methods and linear multistep methods fail to take into account the particular structure of the system, so that they often do not produce satisfactory numerical results. Even some existing geometric integrators such as sym-plectic methods or symmetric methods cannot solve effectively this kind of problem. Therefore, the research of exploring efficient geometric integrators for solving oscilla-tory systems has received more and more attention.Based on the fact stated above, this thesis considers the efficient numerical methods for solving oscillatory systems and combines the methods with Hamiltonian systems. The contents of our work are as follows.Chapter 1 introduces briefly the background of second-order oscillatory systems, the properties of Hamiltonian systems, symplectic and multi-symplectic methods, and energy-preserving methods.Chapter 2 constructs integrators for the multi-frequency oscillatory problems of the formy"(t)+M(t,y(t))y(t)= f(t,y(t)), y(t0)= y0, y’(t0)= y’0. For this kind of system, the classical variation-of-constants approach is not applicable. Although for first-order linear differential equations y’(t)= M(t,y(t)), y(t0)= y0, Magnus method is a choice, however, the procedure involves truncating the expansion of the series which contains many nested commutators first and then approximating the integrals. Even though we apply the Magnus methods to the homogeneous system of oscillatory second-order differential equations, we must transform it into a system of first-order equations with doubled dimension. These make it very complicated and difficult to construct high order practicable algorithms. Therefore, we consider the oscillation of the original system and construct a local equivalent system of nonlinear oscillatory second-order problems with constant matrices to design high-order efficient methods.For perturbed oscillators, two high-efficiency methods are adapted Runge-Kutta-Nystrom (ARKN) and extended Runge-Kutta-Nystrom (ERKN) methods. These two kinds of methods are taking account of the special oscillatory structure of the system and the updates and/or internals are revised. In Chapter 3, we investigate the symplec-ticity and symmetry conditions of ARKN and ERKN methods for separable Hamilto-nian systems. We present the existence of symplectic ARKN methods and show that no ARKN method can be symmetric. Meanwhile, some symplectic and symmetric ERKN methods are derived.Chapter 4 proposes and analyzes a novel fourth-order explicit scheme for solving nonlinear Hamiltonian wave equations with Neumann boundary conditions. First, ac-cording to the characteristic of the wave equation and boundary conditions, a finite difference approximation of order 4 is used for discretizing the second-order spatial derivative and the discrete energy conservation law is established. The semidiscretiza-tion on spartial is of order 4 both at the inner points and on the boundary. Then, a fourth-order multidimensional ERKN method is used for the time integration of the resulting nonlinear oscillatory second-order system of ODEs.Chapter 5 investigates the multi-symplectic integrators for the Hamiltonian wave equations. Taking into account the oscillatory character of Hamiltonian wave equation, we show that the discretization to the Hamiltonian wave equation utilizing two sym-plectic ERKN methods in space and time respectively leads to multi-symplectic inte- grator, and so is the case when a symplectic partitioned Runge-Kutta method is applied in one direction and a symplectic ERKN method in the other. Two multi-symplectic schemes are constructed based on second-order symplectic ERKN methods and the symplectic Stormer-Verlet method. Their prototypes are both the leap-frog scheme.Chapter 6 proposes and analyzes a novel energy-preserving numerical scheme for nonlinear Hamiltonian PDEs based on the blend of spatial discretization by finite ele-ment method and time discretization by the symmetric Average Vector Field method.The matrix-valued φ0(V):=∑k=0∞(-1)kVk/((2k)!)and φ1(V):=Σk=0∞(-1kVk/((2k+1)!)) where V= h2M and h is a stepsize are involved in both ERKN and ARKN methods when solving multidimensional second-order oscillatory system. It is quite difficult to calcu-late efficiently the two functions and the use of the truncation cannot meet the require-ment. In Chapter 7, we present an efficient algorithm to calculate almost exactly the two matrix-valued functions at lower cost.The contributions of this thesis are as follows:● For all the oscillatory systems appear in this thesis, all geometric integrators are analyzed and constructed based on the variation-of-constants formula of (2), thus they make good use of the special structure brought by My and perform well in simulation of the solution of (2).● We consider both symplecticity and symmetry conditions for ARKN and ERKN methods for solving the second-order Hamiltonian system and discuss the ex-istence of the methods. Numerical methods are designed to preserve simulta-neously the symplecticity and symmetry. Moreover, we use symplectic ERKN methods to construct multi-symplectic schemes for solving Hamiltonian PDEs.● By incorporating FEM with the AVF method, an energy-preserving numerical scheme is proposed and analyzed for Hamiltonian wave equations.● For multidimensional systems, all ARKN and ERKN integrators in this thesis use directly the matrix M and avoid the matrix decomposition of M. Meanwhile, we give an algorithm to compute almost exactly the matrix-valued functions φ0(V) and φ1(V) instead of truncating the series simply.