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This paper reviews a particular form of pulsed‐laser‐based thin‐film crystallization method referred to as controlled super‐lateral growth (C‐SLG). By systematically manipulating and controlling the locations, shapes, and extent of melting induced by the incident laser pulses, the C‐SLG approach — notably in a version referred to as sequential lateral solidification (SLS) — can lead to realization of a variety of microstructurally designed crystalline Si films with low structural defect densities, including 1. large‐grained and grain‐boundary‐location controlled polycrystalline films, 2. directionally solidified microstructures, or 3. location‐controlled single‐crystal regions.

In this article, we develop a higher order numerical approximation for time dependent singularly perturbed differential‐difference convection‐diffusion equations. A priori bounds on the exact solution and its derivatives, which are useful for the error analysis of the numerical method are given. We approximate the retarded terms of the model problem using Taylor's series expansion and the resulting time‐dependent singularly perturbed problem is discretized by the implicit Euler scheme on uniform mesh in time direction and a special hybrid finite difference scheme on piecewise uniform Shishkin mesh in spatial direction. We first prove that the proposed numerical discretization is uniformly convergent of O ( Δ t + N − 2 false( ln ⁡ N false) 2 ) , where Δ t and N denote the time step and number of mesh‐intervals in space, respectively. After that we design a Richardson extrapolation scheme to increase the order of convergence in time direction and then the new scheme is proved to be uniformly convergent of O ( Δ t 2 + N − 2 false( ln ⁡ N false) 2 ) . Some numerical tests are performed to illustrate the high‐order accuracy and parameter uniform convergence obtained with the proposed numerical methods.

In this paper, a parameter-uniform fitted mesh finite difference scheme is constructed and analyzed for a class of singularly perturbed interior turning point problems. The solution of this class of turning point problem possess two outflow exponential boundary layers. Parameter-explicit theoretical bounds on the derivatives of the analytical solution are given, which are used in the error analysis of the proposed scheme. The problem is discretized by a hybrid finite difference scheme comprises of midpoint-upwind and central difference operator on an appropriate piecewise-uniform fitted mesh. An error analysis has been carried out for the proposed scheme by splitting the solution into regular and singular components and the method has been shown second order uniform convergent except for a logarithmic factor with respect to the singular perturbation parameter. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.Numerical experiments show that the proposed method gives competitive results in comparison to those of other methods exist in the literature.

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We have developed an efficient two-dimensional numerical model, based on the finite difference method and utilizing the alternate-direction explicit scheme, that can simulate excimer laser melting and solidification of thin Si films on SiO2. The model takes into account important aspects of the process such as undercooling and the temperature-dependent velocity of the solidifying interface, supercooling of liquid Si, and the inert nature of the underlying oxide interface. We demonstrate the unique capability of the model by simulating spatially confined beam-induced localized complete melting of the irradiated portion of the film, and the ensuing lateral solidification, which initiates from the unmelted regions of the film into the completely molten area.

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