Geometric Integrators For Multi-frequency Oscillatory Differential Equations

This thesis concerns the study of geometric integrators for multi-frequency and mul-tidimensional oscillatory second-order ordinary differential equations where M is a d×d positive semi-definite matrix which implicitly contains the frequen-cies of the oscillatory problem (3) and f:Rdâ†'Rd is sufficiently smooth. This system has remarkable structure because of the presence of the linear term Mq. The solution of (3) is a nonlinear multi-frequency oscillator once M is positive semi-definite. These multi-frequency and multidimensional oscillatory problems occur in a wide variety of applications such as physics, astronomy, molecular dynamics, classical and quantum physics, mechanics. Some orbit problems, time-independent Schrodinger equations, semi-discrete wave equations approximated by the method of lines and Fermi-Pasta-Ulam problem all fit the form (3). Since the solution of (3) is a nonlinear multi-frequency oscillator, traditional numerical methods such as Runge-Kutta type methods and linear multistep methods often do not produce satisfactory numerical results. Even some existing geometric integrators such as symplectic methods or symmetric methods cannot effectively solve this kind of problem (3). Therefore, the research of explor-ing efficient geometric integrators for solving multi-frequency oscillatory systems has received more and more attention in recent years.In the past more than ten years, most research is focused on the single-frequency problem where ω>0is the so-called main frequency of the single-frequency problem and ω may be known or accurately estimated in advance. However, multi-frequency and multidimensional oscillatory problems (3) are more complicated coupled systems and the methods for single-frequency problems are not applicable to the multi-frequency oscillatory system (3). The main reasons for this point are as follows. First, the coeffi-cients of the methods for single-frequency problems are functions of v=ωh, however, the M in the multi-frequency oscillatory system (3) is a d x d matrix which implic-itly contains the different frequencies of (3). Thence the methods for single-frequency problems cannot be used directly to the multi-frequency oscillatory system (3). Sec-ond, theoretical analysis of single-frequency problems cannot be extended straightfor-wardly to multi-frequency oscillatory systems since coupled conditions are required in multi-frequency problems. For example, in Chapter3of this thesis, we know that a symplectic multi-frequency and multidimensional method requires more coupled con-ditions than a symplectic single-frequency method. Therefore, the research of solving multi-frequency and multidimensional oscillatory systems (3) is a novel and important challenge.In this thesis we make a systematic investigation of geometric integrators for solving multi-frequency and multidimensional oscillatory second-order ODEs (3). The main contents are organized in Chapters2-8. In Chapter2we analyze and construct effi-cient multi-frequency adapted Runge-Kutta-Nystrom (ARKN) methods. These meth-ods revise the updates of classical Runge-Kutta-Nystrom methods and are constructed based on the variation-of-constants formula of (3), thus they perform well in simula-tion of the solution of (3). In Chapter3we revise not only the updates but also the internal stages of classical Runge-Kutta-Nystrom methods and obtain multi-frequency extended Runge-Kutta-Nystrom (ERKN) methods. In order to study the algebra struc-ture of order conditions for ERKN methods, we establish the set of special extended Nystrom tree (SEN-tree) and define some mappings in the set, based on which the or-der conditions for ERKN methods are derived. Symplecticity conditions and symmetry conditions of multi-frequency ERKN methods are studied and some symplectic and symmetric ERKN methods are derived to preserve simultaneously the symplecticity and symmetry. Based on the schemes of ARKN methods and ERKN methods, Chap-ter4proposes two novel improved Stormer-Verlet formulas with applications to four aspects in scientific computation. In Chapter5we investigate error bounds for explicit ERKN methods and prove that for an important particular case where M is a symmetric and positive semi-definite matrix, the error bound of explicit ERKN methods is inde-pendent of||M||. This result is meaningful especially for the case where||M||≥1. Chapter6formulates a new energy-preserving formula (it is also symmetric) and an-alyzes its properties. In Chapter7we derive trigonometric Fourier collocation (TFC) methods which can be of arbitrary order and this is significant for high accuracy compu-tation. In chapter8we formulate efficient Filon-type asymptotic methods for solving highly oscillatory system (3), where M is a non-singular and diagonalizable matrix having large eigenvalues and||M||≥1.The novelties of this thesis are as follows.First, all the integrators are analyzed and constructed based on the variation-of-constants formula of (3), thus they make good use of the special structure brought by Mq and perform well in simulation of the solution of (3).Second, we derive the symplecticity conditions, symmetry conditions and coupled conditions of ERKN methods for solving multi-frequency and multidimensional oscil-latory Hamiltonian system with the Hamiltonian H(q,p)=1/2pTp+1/2qTMq+U(q).Third, we not only consider methods to preserve respectively symplecticity, sym-metry and Hamiltonian, but also obtain integrators which preserve two properties si-multaneously.Fourth, the procedure of iteration is required for implicit methods. The convergence of traditional energy-preserving methods applied to (3) depends on||M||and this leads to the use of small stepsize. Whereas, it is significant to note that the convergence of energy-preserving methods obtained in this thesis is independent of||M||, which allows us to choose large stepsize in computation. This is a great advantage for long-term integration.Fifth, the well-known Gautschi-type methods for solving (3) and their analysis de-pend on the matrix decomposition of M, whereas all the schemes of geometric integra-tors and their theoretical analysis avoid the matrix decomposition of M and use directly the matrix M. It is known that matrix decomposition usually not only brings additional computation but also results in new errors or other disadvantages especially when the dimension of M is large. Thus the geometric integrators avoiding matrix decomposi-tion in this thesis are more effective and efficient in numerical computations.