The aim of this paper is to design some accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations in rectangular domains. The Laplace transform in time and discrete Fourier transform in space are applied to get the Green's functions of the semi-discretized equations in unbounded domains with single-source. An algorithm is given to compute these Green's functions accurately through some recurrence relations. Furthermore, the finite-difference method is used to discretize the reduced problem with accurate boundary conditions. Numerical simulations are presented to illustrate the accuracy of our method in the case of the linear Schrödinger and heat equations. It is shown that the reflection at the corners is correctly eliminated.

The liquid–gas interface (LGI) on submerged microstructured surfaces has the potential to achieve a large slip effect, which is significant to the underwater applications such as drag reduction. The mechanism of drag reduction in the laminar flow over the LGI has been well recognized, while it is yet not clear for the turbulent boundary layer (TBL) flow over the LGI. In the present work, an experimental system is designed to investigate the mechanism of drag reduction in TBL flow over the LGI. In particular, the flow velocity profile near the LGI is directly measured by high-resolution particle image velocimetry by which the shear stress and the drag reduction are calculated. It is experimentally found that the drag reduction increases as the friction Reynolds number (Reτ0) increases. An analytical expression is derived to analyze the effect of the LGI on drag reduction, which consists of two parts, i.e., the slip property and the modifications to the turbulence structure and dynamics near the LGI. Importantly, the measured slip property also increases as Reτ0 increases, which is demonstrated to be the key effect on drag reduction. This has revealed the mechanism of drag reduction in TBL flow over the LGI. The present work provides physical insights for the drag reduction in TBL flow over the LGI, which is significant to the underwater applications.

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A general method is proposed to build exact artificial boundary conditions for the one-dimensional nonlocal Schrödinger equation. To this end, we first consider the spatial semi-discretization of the nonlocal equation, and then develop an accurate numerical method for computing the Green's function of the semi-discrete nonlocal Schrödinger equation. These Green's functions are next used to build the exact boundary conditions corresponding to the semi-discrete model. Numerical results illustrate the accuracy of the boundary conditions. The methodology can also be applied to other nonlocal models and could be extended to higher dimensions.