A Green's function Monte Carlo algorithm for the Helmholtz equation subject to Neumann and mixed boundary conditions: Validation with an 1D benchmark problem

In this paper, we present the application of our recently developed Green's function Monte Carlo algorithm to the solution of the one-dimensional Helmholtz equation subject to Neumann and mixed boundary conditions problems. The traditional Green's function Monte Carlo approach for the solution of partial differential equations subjected to Neumann and mixed boundary conditions involves “reflecting boundaries” resulting in relatively large computational times. Our algorithm, motivated by the work of K. K. Sabelfeld is philosophically different in that there is no requirement for reflection at these boundaries. The underlying feature of this algorithm is the elimination of the use of reflecting boundaries through the use of novel Green's functions that mimic the boundary conditions of the problem of interest. In the past, we have applied it to the solution of the one-dimensional Laplace equation and the modified Helmholtz equation. In this work, we apply it to the solution of the Helmholtz equation. In the case of the Helmholtz equation, unlike the Laplace equation and modified Helmholtz equation, the algorithm is constrained to quarter-wavelength length scales, a constraint that is the result of resonance in the Green's function for the Helmholtz equation. This constraint is also present in the case of the Helmholtz equation subjected to Dirichlet conditions and is not specific to Neumann and mixed boundary conditions. However, within this constraint, excellent agreement has been obtained between an analytical solution and numerical results.