We have developed a method of solving the Helmholtz equation based on a new way to generalize the “one-way” wave equation, impose correct boundary conditions, and eliminate exponentially growing evanescent waves. The full two-way nature of the Helmholtz equation is included, but the equation is converted into a pseudo one-way form in the framework of a generalized phase-shift structure consisting of two coupled first-order partial differential equations for wave propagation with depth. A new algorithm, based on the particular structure of the coupling between [Formula: see text] and [Formula: see text], is introduced to treat this problem by an explicit approach. More precisely, in a depth-marching strategy, the wave operator is decomposed into the sum of two matrices: The first one is a propagator in a reference velocity medium, whereas the second one is a perturbation term which takes into account the vertical and lateral variation of the velocity. The initial conditions are generated by solving the Lippmann-Schwinger integral equation formally, in a noniterative fashion. The approach corresponds essentially to “factoring out” the physical boundary conditions, thereby converting the inhomogeneous Lippmann-Schwinger integral equation of the second kind into a Volterra integral equation of the second kind. This procedure supplies artificial boundary conditions, along with a rigorous method for converting these solutions to those satisfying the correct, Lippmann-Scwinger (physical) boundary conditions. To make the solution numerically stable, the Feshbach projection operator technique is used to remove only the nonphysical exponentially growing evanescent waves, while retaining the exponentially decaying evanescent waves, along with all propagating waves. Suitable absorbing boundary conditions are implemented to deal with reflection or wraparound from boundaries. At the end, the Lippmann-Schwinger solutions are superposed to produce time snapshots of the propagating wave.
Asymptotic stability of exogenous chemotaxis systems with physical boundary conditions