On the numerical solution of the iterative parabolic equations by ETDRK pseudospectral methods in linear and nonlinear media

Abstract

Iterative parabolic equations (IPEs) were recently introduced as a promising tool for solving wave propagation problems in complex linear and non-linear media. In this study a general approach to the numerical solution of iterative parabolic equations on the basis of powerful ETD pseudospectral method is developed. The solution of nth IPE requires the computation of the input term that is obtained by differentiation of the solution of n−1th IPE. The numerical noise resulting from such differentiation spoils the solutions of higher-order IPEs. It is shown that this noise can be suppressed via a filtering procedure involving Chebyshev polynomials of a discrete variable. In this study Padé type iterative parabolic approximations are introduced both for the case of linear and nonlinear media. It is shown that Padé-type approximations are more suitable for solving the problems of wave propagation.