Structure-Preserving Algorithms For Highly Oscillatory Systems

A large number of problems arising in astronomy, quantum mechanics, biological chemistry, molecular dynamics and so on can be modeled by systems of highly oscillatory differential equations. With their complexity and significance, the design and analysis of numerical methods for solving highly oscillatory problems has been a difficult but attracting field of numerical analysis of differential equations.Now theories and methods in the solution of highly oscillatory differential equations have been abundant. There have been the asymptotic method and Filon-type methods for the integrals of highly oscillatory function; for second-order highly oscillatory differential equations, the approach of adiabatic transform followed by numerical approximation is proved to be superior to direct attack. For the preservation of the total energy and the oscillatory energies of the exact and numerical solutions of highly oscillatory Hamilton systems, the traditional backward error analysis loses effect. As recipe, a new analytical tool, called the modulated Fourier expansion, emerged. The focus of our work lies on proposing more effective Filon-type methods for systems of second-order oscillatory differential equations and analyzing the almost preservation of energy of the exact and numerical solutions of multi-frequencies Hamiltonian problems in a general form.This thesis is divided into four chapters.Chapter1surveys the asymptotic and Filon-type methods for highly oscillatory integrals, the combination of waveform relaxation and Filon-type methods for systems of nonlinear second order highly oscillatory differential equations and in addition we introduce the adiabatic and Gautschi-type methods. Also the basic theory of modulated Fourier expansion is briefly presented.In Chapter2we propose new Filon-type numerical methods for second order highly oscillatory differential equations. In order to take full advantages of the adiabatic and Filon-type methods, we first transform the problems into adiabatic variables and then solve the resulted equations with Filon-type methods. We present the Adiabatic Filon-type methods for linear systems (AFL) and then the adiabatic wave relaxation Filon-type (AWRF) methods for nonlinear systems. The numerical results produced by these methods applied to some typical test problems show that our methods are more effective than the existing ones in the literature.In chapter3, for the inhomogeneous second order highly oscillatory differential equations with time-varying frequencies, we propose the adiabatic integrators method (FAD) which transforms the original problem into a system of first order differential equations whose solution is an adiabatic invariant, and then solves the system by the variation-of-constants approach. Numerical experiment illustrates that the effectiveness and competence of the new method.In chapter4, we discuss numerical solution of non-resonant multi-frequencies Hamilton systems in a general form. Employing the modulated Fourier expansion we prove that the total and oscillatory energies of exact and numerical solutions are almost-invariant. The numerical experiments show that our method almost preserves the structure of this category of Hamiltonian problems.Finally, we summarize the main conclusions of this thesis and list some future prospects which challenge our work on highly oscillatory problems.