Transparent boundary conditions for iterative high-order parabolic equations

Recently a new approach to the construction of high-order parabolic approximations for the Helmholtz equation was developed. These approximations have the form of the system of iterative parabolic equations, where the solution of the n  -th equation is used as an input term for the (n+1)(n+1)-th equation. In this study the transparent boundary conditions for such systems of coupled parabolic equations are derived. The existence and uniqueness of the solution of the initial boundary value problem for the system of iterative parabolic equations with the derived boundary conditions are proved. The well-posedness of this problem is also established and an unconditionally stable finite difference scheme for its solution is proposed.