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

2013. No. 1309.7280.

Ducomet B., Злотник А. А., Романова А. В.

An initial-boundary value problem for the $n$-dimensional ($n\geq 2$) time-dependent Schrödinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for $n\geq 3$. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting method, the uniqueness of solution and the unconditional uniform in time $L^2$-stability (in particular, $L^2$-conservativeness) are proved. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of theparallelepiped) is applicable for general potential. Numerical results on the 2D tunnel effect for a Pöschl-Teller-like potential-barrier and a rectangular potential-well are also included.

Ключевые слова: устойчивость, discrete transparent boundary conditions, stability, дискретные прозрачные граничные условия, схема, туннельный эффект, нестационарное уравнение Шрёдингера, the Crank-Nicolson finite-difference scheme, The Strang splitting, расщепление Стренга, tunnel effect, the time-dependent Schrodinger equation, дискретизация Кранк-Никольсон по времени, higher-order scheme, разностная схема повышенного порядка