Propagation in Helmholtz Waveguides Using Dtn and Ntd Maps

For many wave propagation problems, it is often necessary to solve the variable coeecient Helmholtz equation in a very large domain. The one-way re-formulation based on the Dirichlet-to-Neumann (DtN) map can be used to nd the solutions of the Helmholtz equation over a large range distance. It is particularly useful for waveguide problems, since it leads to a numerical algorithm that marches in the range variable. The DtN map satisses an operator Riccati equation that may blow up even though the original problem is well-posed. In this paper, we develop a method that automatically switches between the DtN and Neumann-to-Dirichlet (NtD) maps, to avoid the singularities of these operators. When implemented with a truncated local eigenfunction expansion, the operator equations can be eeciently solved.