Structure-Preserving Algorithms for Oscillatory Differential Equations

Book Preface

Effective numerical solution of differential equations, although as old as differential equations themselves, has been a great challenge to numerical analysts, scientists and engineers for centuries. In recent decades, it has been universally acknowledged that differential equations arising in science and engineering often have certain structures that require preservation by the numerical integrators. Beginning with the symplectic integration of R. de Vogelaere (1956), R.D. Ruth (1983), Feng Kang (1985), J.M. Sanz-Serna (1988), E. Hairer (1994) and others, structure-preserving computation, or geometric numerical integration, has become one of the central fields of numerical differential equations. Geometric numerical integration aims at the preservation of the physical or geometric features of the exact flow of the system in long-term computation, such as the symplectic structure of Hamiltonian systems, energy and momentum of dynamical systems, time-reversibility of conservative mechanical systems, oscillatory and high oscillatory systems.

The objective of this monograph is to study structure-preserving algorithms for oscillatory problems that arise in a wide range of fields such as astronomy, molecular dynamics, classical mechanics, quantum mechanics, chemistry, biology and engineering. Such problems can often be modeled by initial value problems of secondorder differential equations with a linear term characterizing the oscillatory structure of the systems. Since general-purpose high order Runge–Kutta (RK) methods, Runge–Kutta–Nyström (RKN) methods, and linear multistep methods (LMM) cannot respect the special structures of oscillatory problems in long-term integration, innovative integrators have to be designed. This monograph systematically develops theories and methods for solving second-order differential equations with oscillatory solutions.

As the basis of the whole monograph, Chap. 1 reviews the general notions and ideas related to the numerical integration of oscillatory differential equations. Chapter 2 presents multidimensional RKN methods adapted to second-order oscillatory systems.