A coupled marching method for Cauchy problems of the Helmholtz equation in complex waveguides

This paper develops a coupled marching method to solve Cauchy problems of the Helmholtz equation in waveguides with irregularly varying local parts. We reformulate the spectral projection marching method (SPMM) to act as a coupler to couple itself with the operator marching method (OMM). In marching computations, the coupled marching method applies the SPMM to compute wave propagations in irregular areas, and applies the OMM to slowly varying parts. We also present transformation error estimates for analyzing errors arising in the coordinate transformation between different local subspaces, and discuss principles for applying the coupled marching method. Extensive numerical experiments demonstrate the accuracy, convergence and efficiency of the coupled marching method in various complex waveguides.