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

Abstract

MSC: 65M06 65M50 65Z05 78A15 78M20 Keywords:Splitting method Transformations Linear and nonlinear equations Highly oscillatory wave problems Asymptotic stability Algorithmic efficiency a b s t r a c t 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.

Exponential splitting for n-dimensional paraxial Helmholtz equation with high wavenumbers

“…Though amplitudes of perturbations are relatively small as compared with focusing peaks, wave oscillations are still persistent and clearly visible throughout simulation experiments. The phenomenon well agrees with explorations of errors in linearized and localized wave approximations [15,16].…”

“…It is evident that, when κ 0 ≫ 10, the field function E becomes highly oscillatory and continuing refinements of the computational mesh over D are required for maintaining acceptable computational accuracy [14]. This may not only lead to unrealistically small step sizes, but also impair the overall computational efficiency even when standard exponential splitting procedures are deployed [3,15,12]. It has been therefore meaningful to improve conventionally used exponential splitting strategies for overcoming above-mentioned difficulties.…”

“…This makes fast numerical solutions of (2.1) to be realistic. In fact, computational results from exponential splitting schemes or spectral approximations based on (2.2) have demonstrated its huge potential, though the study of its numerical stability is still in its infancy, partially due to relatively complex operator structures of (1.6)-(1.8) [8,15,14].…”

“…(4.1), (4.2) model a variety of highly oscillatory optical waves [11]. Any reliable numerical method for solving the problem must be asymptotically stable and reproduce correctly the underlying focusing phenomenon [11,15,12]. This has become an effective testing stone for the correctness and accuracy of highly oscillatory wave simulations.…”

On Classification of Symmetry Reductions for the Eikonal Equation

“…Sheng, Guha and Gonzalez (see [8,9] and the references therein) proposed new effective methods for solving many problems of computational optics.…”