The Riccati method for the Helmholtz equation

The operator Riccati equation for the Dirichlet-to-Neumann map is derived from the exact operator factorization of the two-dimensional variable coefficient Helmholtz equation. Numerical schemes are developed for the operator Riccati equation and its variant using a local eigenfunction expansion. This leads to a practical computational method for acoustic wave propagation over large range distances, since the boundary value problem of the Helmholtz equation is reduced to 'initial' value problems that are solved by marching in the range. The efficiency and accuracy of the method is demonstrated by numerical experiments including the plane-parallel range- dependent waveguide benchmark problem proposed by the Acoustical Society of America.