Discrete Transparent Boundary Conditions for General Schrödinger–Type Equations

Transparent boundary conditions (TBCs) for general Schrödinger– type equations on a bounded domain can be derived explicitly under the assumption that the given potential V is constant on the exterior of that domain. In 1D these boundary conditions are non–local in time (of memory type). Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nicolson finite difference method only conditionally stable. In this paper a novel discrete TBC is derived directly from the discrete whole–space problem that yields an unconditionally stable scheme. Numerical examples illustrate the superiority of the discrete TBC over other existing consistent discretizations of the differential TBCs. As an application of these boundary conditions to wave propagation problems in underwater acoustics results for the so–called standard and wide angle “parabolic” equation (SPE, WAPE) models are presented.