Explicit Gautschi-type integrators for nonlinear multi-frequency oscillatory second-order initial value problems

• Published in: Numerical algorithms Vol. 81; no. 4; pp. 1275 - 1294
• Main Authors: Shi, Wei, Wu, Xinyuan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2019; Springer Nature B.V
• Subjects: Algebra; Algorithms; Applied mathematics; Approximation; Boundary value problems; Computer Science; Error analysis; Integrators; Mathematical analysis; Numeric Computing; Numerical Analysis; Numerical methods; Ordinary differential equations; Original Paper; Theory of Computation; 65L05; 65L06; 34C15; Error analysis; Structure preservation; Kepler’s problem; 65F30; Gautschi-type integrators; Oscillatory nonlinear second-order initial value problems; Oscillatory Hamiltonian systems
• DOI: 10.1007/s11075-018-0635-7
• ISSN: 1017-1398

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.