Explicit Gautschi-type integrators for nonlinear multi-frequency oscillatory second-order initial value problems - Shi, Wei; Wu, Xinyuan (Numerical algorithms, 2019)
The main theme of this paper is explicit Gautschi-type integrators for the nonlinear multi-frequency oscillatory second-order initial value problems of the form y ′′ = − A ( t , y ) y + f ( t , y ) , y ( t 0 ) = y 0 , y ′ ( t 0 ) = y 0 ′ . This work is important and interesting within the broader framework of the subject. In fact, the Gautschi-type methods for oscillatory problems with a constant matrix A have been investigated by many authors. The key question now is that the classical variation-of-constants approach is not applicable to the oscillatory nonlinear problems with a variable coefficient matrix A ( t , y ). We consider successive approximations or locally equivalent systems for the problems, and derive efficient explicit Gautschi-type integrators. The error analysis is presented for the local approximation accordingly. Accompanying numerical results demonstrate the remarkable efficiency of the new Gautschi-type integrators in comparison with some existing numerical methods in the scientific literature.