High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering

When solving linear scattering problems, one typically first solves for the impinging wave in the absence of obstacles. Then, using the linear superpo sition principle, the original problem is reduced to one which involves onl y the scattered wave (which is driven by the values of the impinging field at the surface of the obstacles). When the original domain is unbounded, sp ecial artificial boundary conditions (ABCs) have to be set at the outer (ar tificial) boundary of the finite computational domain in order to guarantee the reflectionless propagation of waves through this external artificial b oundary. The situation becomes conceptually different when the propagation equation is nonlinear. In this case the impinging and scattered waves can n o longer be separated, and the problem has to be solved in its entirety. In particular, the boundary on which the incoming field values are prescribed should transmit the given incoming waves in one direction and simultaneous ly be transparent to all the outgoing waves that travel in the opposite dir ection. We call such boundary conditions two-way ABCs. In the paper, we con struct the two-way ABCs for the nonlinear Helmholtz equation, which models a continuous-wave laser hewn propagation in a medium with a Kerr nonlinear index of refraction. In this case, the forward propagation of the beam is a ccompanied by backscattering, i.e., generation of waves in the opposite dir ection to that of the incoming signal. Our two-way ABCs generate no reflect ion of the backscattered waves and at the same time impose the correct valu es of the incoming wave. The ABCs are obtained in the framework of a fourth -order accurate discretization to the Helmholtz operator inside the computa tional domain. The fourth-order convergence of our methodology is corrobora ted experimentally by solving linear model problems. We also present soluti ons in the nonlinear case using the two-way ABC which, unlike the tradition al Dirichlet boundary condition approach, allows for direct calculation of the magnitude of backscattering. (C) 2001 Academic Press.