High-Order Two-Way Artificial Boundary Conditions for Nonlinear Wave Propagation with Backscattering

Abstract

When solvinglinear scattering problems, one typically first solves for the impinging wave in the absence of obstacles. Then, using the linear superposition principle, the original problem is reduced to one which involves only 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, special artificial boundary conditions (ABCs) have to be set at the outer (artificial) boundary of the finite computational domain in order to guarantee the reflectionless propagation of waves through this external artificial boundary. The situation becomes conceptually different when the propagation equation is nonlinear. In this case the impinging and scattered waves can no 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 simultaneously be transparent to all the outgoing waves that travel in the opposite direction. We call such boundary conditions two-way ABCs. In the paper, we construct the two-way ABCs for the nonlinear Helmholtz equation, which models a continuous-wave laser beam propagation in a medium with a Kerr nonlinear index of refraction. In this case, the forward propagation of the beam is accompanied by backscattering, i.e., generation of waves in the opposite direction to that of the incoming signal. Our two-way ABCs generate no reflection of the backscattered waves and at the same time impose the correct values of the incoming wave. The ABCs are obtained in the framework of a fourth-order accurate discretization to the Helmholtz operator inside the computational domain. The fourth-order convergence of our methodology is corroborated experimentally by solving linear model problems. We also present solutions in the nonlinear case using the two-way ABC which, unlike the traditional Dirichlet boundary condition approach, allows for direct calculation of the magnitude of backscattering.