Discrete artificial boundary conditions

Abstract

When computing numerically the solution of a partial differential equation in an unbounded domain usually artificial boundaries are introduced to limit the computational domain. Special boundary conditions are derived at this artificial boundaries to approximate the exact whole– space solution. If the solution of the problem on the bounded domain is equal to the whole– space solution (restricted to the computational domain) these boundary conditions are called transparent boundary conditions (TBCs). This dissertation is concerned with transparent boundary conditions for convection–diffusion equations and general Schrodinger–type pseudo–differential equations arising from “parabolic” equation (PE) models which have been widely used for one–way wave propagation problems in various application areas, e.g. seismology, optics and plasma physics. As a special case the Schrodinger equation of quantum mechanics is included. Existing discretizations of these TBCs induce numerical reflections at this artificial boundary and also may destroy the stability of the used finite difference method. To overcome both problems we propose a new discrete TBC which is derived from the fully discretized whole– space problem. This discrete TBC is reflection–free and conserves the stability properties of the whole–space scheme. While we shall assume a uniform discretization in time, the interior spatial discretization may be nonuniform. The superiority of the new discrete TBC over existing discretizations is illustrated on several benchmark problems.