Discrete non–local boundary conditions for Split–Step Padé Approximations of the One–Way Helmholtz Equation

This paper deals with the efficient numerical solution of the two–dimensional one– way Helmholtz equation posed on an unbounded domain. In this case one has to introduce artificial boundary conditions to confine the computational domain. The main topic of this work is the construction of so–called discrete transparent boundary conditions for state-of-the-art parabolic equations methods, namely a split–step discretization of the high–order parabolic approximation and the split–step Padé algorithm of Collins. Finally, several numerical examples arising in optics and underwater acoustics illustrate the efficiency and accuracy of our approach.