We propose a way to efficiently treat the well-known transparent boundary conditions for the Schrödinger equation. Our approach is based on two ideas: to write out a discrete transparent boundary condition (DTBC) using the Crank-Nicolson finite difference scheme for the governing equation, and to approximate the discrete convolution kernel of DTBC by sum-of-exponentials for a rapid recursive calculation of the convolution.We prove stability of the resulting initial-boundary value scheme, give error estimates for the considered approximation of the boundary condition, and illustrate the efficiency of the proposed method on several examples.

For an external problem in IRd (d=2, 3) such that the unknown function satisfies the wave equation outside a finite domain, we generate artificial boundary conditions transparent to outgoing waves. These conditions permit an equivalent replacement of the original external problem by the problem inside the artificial boundary which is a circle (d=2) or a sphere (d=3): The questions of numerical implementation of the artificial conditions (that are non-local in both space and time) are considered. Special attention is paid to the reduction of necessary computational resources; in particular, a way of incorporating these conditions into numerical methods which makes the computational formulae local in time is suggested. The aspects of treating artificial boundaries of a non-spherical shape are discussed. Numerical examples of two- and three-dimensional scattering problems demonstrate the accuracy of proposed artificial boundary conditions.

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We consider a time-dependent flow problem in an infinitely long wind tunnel of circular cross-section. It is assumed that beginning with a distance up-and downstream from the stream-lined body, the flow is approximately described by the time-dependent Euler equations linearized about a uniform free-stream flow. By using this linear model, we transfer the boundary conditions from infinity to the front and back boundaries of a finite computational domain. The obtained bound-Ž . ary conditions called transparent ones TBCs are represented by explicit formulae in the physical space. The TBCs contain operators of Fourier expansion with respect to eigenfunctions of the cross-section and convolutions with respect to time, i.e., they are non-local in both space and time. One possibility of their numerical implementation is described; the main attention is paid to the evaluation of the convolution operator. Due to the representation of the convolution kernel by an exponential polynomial, the evaluation formulae become even local in time.

We propose highly accurate finite-difference schemes for simulating wave propagation problems described by linear second-order hyperbolic equations. The schemes are based on the summation by parts (SBP) approach modified for applications with violation of input data smoothness. In particular, we derive and implement stable schemes for solving elastodynamic anisotropic problems described by the Navier wave equation in complex geometry. To enhance potential of the method, we use a general type of coordinate transformation and multiblock grids. We also show that the conventional spectral element method (SEM) can be treated as the multiblock finite-difference method whose blocks are the SEM cells with SBP operators on GLL grid.