Nonreflecting Boundary Condition for the free Schrödinger equation for hyperrectangular computational domains

Abstract

In this article, we discuss the efficient ways of implementing the transparent boundary condition (TBC) and its various approximations for the free Schrödinger equation on a hyperrectangular computational domain in $\field{R}^d$ with periodic boundary conditions along the $(d-1)$ unbounded directions. In particular, we consider Padé approximant based rational approximation of the exact TBC and a spatially local form of the exact TBC obtained under its high-frequency approximation. For the spatial discretization, we use a Legendre-Galerkin spectral method with a boundary-adapted basis to ensure the bandedness of the resulting linear system. Temporal discretization is then addressed with two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). Finally, several numerical tests are presented to demonstrate the effectiveness of the methods where we study the stability and convergence behaviour empirically.

Figures (9)

• Figure 1: The figure shows a rectangular domain with periodic boundary conditions along vertical axis.
• Figure 2: A schematic is shown in this figure which depicts the evolution of the auxiliary field $\varphi(x_1,x_2,\tau_1,\tau_2)$ in the $(\tau_1,\tau_2)$-plane starting from the diagonal which also serves as initial conditions for solving the IVPs (arrow in the line depicts the evolution direction in time). Note that TBCs on $\partial{\Omega_i}$ require the history of the auxiliary field along the horizontal line up to the diagonal in the $(\tau_1,\tau_2)$-plane.
• Figure 3: A schematic depiction of the evolution of the auxiliary fields $\varphi_{k,a_1}(x_2,\tau_1,\tau_2)$ in the $(\tau_1,\tau_2)$-plane is provided in this figure.
• Figure 4: The figure shows a rectangular cuboidal domain with boundary faces parallel to one of the axes.
• Figure 5: A schematic is shown in this figure which depicts the evolution of the auxiliary field $\varphi(x_1,\boldsymbol{x}_{\perp},\tau_1,\tau_{\perp})$ in the $(\tau_1,\tau_{\perp})$-plane starting from the diagonal which also serves as initial conditions for solving the IVPs (arrow in the line depicts the evolution direction in time). Note that TBCs on $\partial{\Omega_i}$ require the history of the auxiliary field along the horizontal line up to the diagonal in the $(\tau_1,\tau_{\perp})$-plane.

Theorems & Definitions (1)

• Remark 1