Geometric Numerical Integration For Second-order Oscillatory Conservative/Dissipative Systems

This thesis is concerned with numerical integration of multi-frequency and multidi-mensional oscillatory second-order differential equations where M∈Rd×d is a positive semidefinite constant real matrix containing implicitly the frequencies of the problem, y ∈Rd and f:Rd×Rdâ†'Rd.When the right-hand-side function f in (2) depends on y’, the system is damped, and in this case, the energy of the system is dissipative. If M is a symmetric and positive semi-definite matrix and f(y) is the negative gradient of a real-valued function U(y) in (2), then the corresponding system is conservative that can be described by an oscilla-tory Hamiltonian system, which has two important properties:energy conservation and symplecticity.It has now become a common practice that in the design of numerical algorithms, the special structure of the underlying problem should be considered. In recent years, geometric numerical integration has drawn much attentions. A numerical integration method is called geometric if it exactly preserves one or more physical/geometric prop-erties of the system. More precisely, to make numerical methods correctly reproduce the qualitative behaviour of the systems, one should try to ensure that the basic struc-tures of the system is not to be altered by the numerical methods. Therefore, our atten-tion has been focused on geometric numerical integrators for system (2) that preserve as many structures as possible. A pronounced structure of the system (2):oscillation is always considered throughout the thesis.The first aspect of this thesis is concerned with the oscillatory dissipative sys-tem. We present the analysis of the concrete multidimensional adapted Runge-Lutta-Nystrom (ARKN) methods for damped (involving y’) multidimensional oscillatory second-order initial value problems. Meanwhile, by introducing a novel linear test model, the phase properties and stability of numerical methods for general oscillatory second-order initial value problems whose right-hand-side functions depend on both y and y’ are analyzed, the new definitions of dispersion and dissipation are given which can be viewed as an extension of the traditional ones. They are more suitable for nu-merical methods designed for the general second-order initial value problem involving both y and y’.The second aspect is concerned with oscillatory conservative system. Firstly, by further discussion of the symplecticity and symmetry of ARKN and extended Runge-Kutta-Nystrom (ERKN) integrators, high-order symplectic and symmetric composition methods based on the ARKN integrators and ERKN integrators are constructed for os-cillatory Hamiltonian system with Hamiltonian Secondly, in terms of energy conservation, incorporating the idea of the discrete gradi-ent method into the ERKN integrator, we present an extended discrete gradient formula for the oscillatory Hamiltonian system. The new formula can preserve energy exactly as well as adapt to the oscillatory structure of the system. The convergence is analyzed for the implicit schemes based on the discrete gradient formula. It shows that the con-vergence of the new formula is independent of the matrix M and is much faster than the traditional discrete gradient formula. Furthermore, we apply the extended discrete gra-dient formula to conservative nonlinear wave partial differential equations (PDEs) and present a linearly-fitted conservative scheme numerically solving conservative nonlin-ear wave PDEs. After a minor modification to the extended discrete gradient formula, a linearly-fitted scheme which can preserve the dissipation of energy is applicable to dissipative wave PDEs.