On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations
The paper deals with the numerical analysis of highly oscillatory second-order differential equations with dominating linear term. Numerical standard schemes applied to such an equation are forced to use step sizes \(h\) smaller than the square root of the linear term. Using an appropriate formal representation of the exact solution the author derives a numerical scheme with exponential Gautschi-type integrators. For solutions satisfying a finite energy condition second-order convergence of the numerical scheme is shown. Here the constant is independent of the norm of the dominating linear term, that is, independent of the frequency of the exact solution. Integrators of that type are particularly well suited for the long-time integration of oscillatory second order differential equations. Finally, a numerical example from molecular dynamics is presented.
scientific article; zbMATH DE number 2156012