Abstract: 

 

This talk concerns the asymptotic stability of an eikonal, or ray, transformation based Peaceman-Rachford splitting method for solving the paraxial Helmholtz equation with high wave numbers. Arbitrary nonuniform grids are considered in transverse and beam propagation directions. The differential equation targeted has been used for modeling propagations of high intensity laser pulses over a long distance without diffractions. Self-focusing of high intensity beams may be balanced with the de-focusing effect of created ionized plasma channel in the situation, and applications of grid adaptations are frequently essential. It is shown that the oscillation-free decomposition method on arbitrary adaptive grids is asymptotically stable with a stability index one. The result can be extended to multidimensional cases for multiphysical high-oscillatory applications. Simulation experiments are given to illustrate our concern and conclusions.