The Numerov-Crank-Nicolson Scheme on a Non-Uniform Mesh for the Time-Dependent Schrödinger Equation on the Half-Axis

Kinetic and Related Models. 2015. Vol. 8. No. 3. P. 587–613.

Zlotnik Alexander

We deal with the initial-boundary value problem for the 1D time-dependent Schrödinger equation on the half-axis. The scheme with the Numerov averages on the non-uniform space mesh and of the Crank-Nicolson type in time is studied, with some approximate transparent boundary conditions (TBCs). Deriving bounds for the skew-Hermitian parts of the Numerov sesquilinear forms, we prove the uniform in time stability in $L^2$- and $H^1$-like space norms under suitable conditions on the potential and the meshes. In the case of the discrete TBC, we also derive higher order in space error estimates in both norms in dependence with the Sobolev regularity of the initial function (and the potential) and properties of the space mesh. Numerical results are presented for tunneling through smooth and rectangular potentials-wells, including the global Richardson extrapolation in time to ensure also higher order in time.

Ключевые слова: устойчивость, The time-dependent Schrödinger equation, approximate and discrete transparent boundary conditions, приближенные и дискретные прозрачные граничные условия, unbounded domains, неограниченные области, нестационарное уравнение Шрёдингера, the Numerov discretization in space, the Crank-Nicolson discretization in time, дискретизация Нумерова по пространству, дискретизация Кранк-Никольсон по времени, stability, the Richardson extrapolation, экстраполяция Ричардсона, оценки погрешности, error estimates