Family of finite-difference schemes with transparent boundary conditions for the nonstationary Schrödinger equation in a semi-infinite strip

On a splitting higher-order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped

“…In this paper, in the spirit of [21,22], we first split these operators (in space) and recover the properties without reducing the higher order, for any n 2. We second apply the Strang-type splitting in potential in time also conserving the higher order; the resulting scheme can be called "double-(space-time)-splitting".…”

“…One of them exploits the so-called discrete transparent boundary conditions (TBCs) at artificial boundaries [3,11]. Its advantages are the complete absence of spurious reflections in practice as well as the rigorous mathematical background and stability results in theory.The discrete TBCs for the Crank-Nicolson finite-difference scheme, the higher order Numerov-Crank-Nicolson scheme and a general family of schemes on an infinite or semiinfinite strip were constructed and studied respectively in [3,7,8], [17] and [21,22]. All these schemes are implicit, so to implement them, solving of specific complex systems of linear algebraic equations is required at each time level.The splitting technique is widely used to simplify numerical solving of the timedependent Schrödinger and related equations, in particular, see [4,5,13,14,15,19].…”

“…The known Strang-type splitting with respect to the potential has been recently applied to the Crank-Nicolson and the Numerov-Crank-Nicolson scheme with the discrete TBCs in 2D case in [10,20].Higher order methods are important due to their ability to reduce computational costs essentially, and the Numerov-Crank-Nicolson scheme can be written in n-dimensional case as well. But we show that, for n 3, the Numerov space operators lose their important spectral properties existing for n = 2 so that the scheme becomes impractical.In this paper, in the spirit of [21,22], we first split these operators (in space) and recover the properties without reducing the higher order, for any n 2. We second apply the Strang-type splitting in potential in time also conserving the higher order; the resulting scheme can be called "double-(space-time)-splitting".…”

“…For this scheme on an infinite space mesh in the semi-infinite parallelepiped, we prove the unconditional uniform in time L 2stability together with the mass conservation law using combination of techniques from [9,10,17].The discrete TBCs allow to restrict rigorously solutions of the schemes on infinite space meshes to finite ones; they can be written in several forms. Our form of the discrete TBC and its derivation for the "double-splitting" scheme follow [7,9,21,22]. This form simplifies the whole study and is computationally stable.…”

Error Estimates of the Crank-Nicolson-Polylinear FEM with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped

“…They first were constructed and studied for the standard Crank-Nicolson in time finite-difference schemes, see [1,5] and also [2,3], in the cases of the infinite or semi-infinite axis and strip. Later families of finitedifference schemes with general space averages were treated in [4,12,15]. In particular, they include the linear and bilinear FEMs in space.…”