An exponential transformation based splitting method for fast computations of highly oscillatory solutions

Splitting, or decomposition, methods have been widely used for achieving higher computational efficiency in solving wave equations. A major concern has remained, however, if the wave number involved is exceptionally large. In the case, merits of a conventional splitting method may diminish due to the fact that tiny discretization steps need to be employed to compensate high oscillations. This paper studies an alternative way for solving highly oscillatory paraxial wave problems via a modified splitting strategy. In the process, an exponential transformation is first introduced to convert the underlying differential equation to coupled nonlinear equations. Then the resulted oscillation-free system is treated by a Local-One-Dimensional (LOD) scheme for desired accuracy, efficiency and computability. The splitting method acquired is asymptotically stable and easy to use. Computational experiments are given to illustrate our numerical procedures.