Discrete transparent boundary conditions for parabolic equations

Abstract

There are simple algorithms for constructing transparent boundary conditions (TBC's) for a partial discretization of the basic parabolic equation that is known as a "semi-discrete" parabolic equation. This equation and some of these algorithms are reviewed. Solutions of a semi-discrete parabolic equation in a long rectangular strip subject to TBC's at the long edges of the strip are then considered. These solutions can be computed accurately and efficiently with a pseudospectral method that is based on expansions in Chebyshev polynomials. It is beneficial to combine this method with a conventional split-step FFT solution of a parabolic equation subject to Neumann boundary conditions at the long edges of the strip. This hybrid approach will be called the "decomposition method". It is demonstrated in a computation of radiation modes from the termination of a truncated nonlinear internal gravity wave duct in a shallow ocean area.