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Think globally, act locally: Solving highly-oscillatory ordinary differential equations. (English) Zbl 1016.65050
In a number of applications we are faced with the problem of finding a numerical solution of an ordinary differential equation whose exact solution is strongly oscillatory. For problems of this type, the author first discusses the issue of error propagation. Special attention is paid to Runge-Kutta and Magnus methods. As a result of the analysis, a modified Magnus method is constructed that exhibits a very good performance.
MSC:
65L06Multistep, Runge-Kutta, and extrapolation methods
34A34Nonlinear ODE and systems, general
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
65L05Initial value problems for ODE (numerical methods)
34A26Geometric methods in differential equations
65Y20Complexity and performance of numerical algorithms