Conventional and Asymptotic Stabilities of Decomposed Compact Methods for Solving Highly Oscillatory Wave Problems
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Abstract
Radially symmetric transverse fields and standard polar coordinates are considered. A decomposition is implemented to remove the anticipated singularity in the transverse direction. Compact scheme structures are introduced in the transverse direction to raise the accuracy and efficiency of the scheme developed for laser and subwavelength optical computations. While the highly accurate compact algorithm shies away from the conventional stability, it is shown to be asymptotically stable with index one. This physically relevant asymptotic stability facilitates the algorithmic effectiveness, reliability, and applicability. Numerical experiments further demonstrate the algorithm's high reliability when implemented in highly oscillatory optical self-focusing optical beam propagation simulations. Computational examples are presented to illustrate the conclusions.