18 research outputs found
Cost-reduction implicit exponential Runge-Kutta methods for highly oscillatory systems
In this paper, two novel classes of implicit exponential Runge-Kutta (ERK)
methods are studied for solving highly oscillatory systems. First of all, we
analyze the symplectic conditions of two kinds of exponential integrators, and
present a first-order symplectic method. In order to solve highly oscillatory
problems, the highly accurate implicit ERK integrators (up to order four) are
formulated by comparing the Taylor expansions of numerical and exact solutions,
it is shown that the order conditions of two new kinds of exponential methods
are identical to the order conditions of classical Runge-Kutta (RK) methods.
Moreover, we investigate the linear stability properties of these exponential
methods. Finally, numerical results not only present the long time energy
preservation of the first-order symplectic method, but also illustrate the
accuracy and efficiency of these formulated methods in comparison with standard
ERK methods