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Operator splitting methods have been applied to nonlinear partial differential equations that involve operators of different nature. The main idea of these methods is to decompose a complex equation into simpler sub-equations, which can be solved separately. The main advantage of the operator splitting methods is that they provide a great flexibility in choosing different numerical methods, depending on the feature of each sub-problem. In this dissertation, we have developed highly accurate and efficient numerical methods for several nonlinear partial differential equations, which involve both linear and nonlinear operators. We first propose a fast explicit operator splitting method for the modified Buckley-Leverett equations which include a third-order mixed derivatives term resulting from the dynamic effects in the pressure difference between the two phases. The method splits the original equation into two equations, one with a nonlinear convective term and the other one with high-order linear terms so that appropriate numerical methods can be applied to each of the split equations: The high-order linear equation is numerically solved using a pseudo-spectral method, while the nonlinear convective equation is integrated using the Godunov-type central-upwind scheme. The spatial order of the central-upwind scheme depends on the order of the piecewise polynomial reconstruction: We test both the second-order minmod-based reconstruction and fifth-order WENO5 one to demonstrate that using higher-order spatial reconstruction leads to more accurate approximation of solutions. We then propose fast and stable explicit operator splitting methods for two phase-field models (the molecular beam epitaxy equation with slope selection and the Cahn-Hilliard equation), numerical simulations of which require long time computations. The equations are split into nonlinear and linear parts. The nonlinear part is solved using a method of lines combined with an efficient large stability domain explicit ODE solver. The linear part is solved by a pseudo-spectral method, which is based on the exact solution and thus has no stability restriction on the time step size. We have verified the numerical accuracy of the proposed methods and demonstrated their performance on extensive one- and two-dimensional numerical examples, where different solution profiles can be clearly observed and are consistent with previous analytical studies. / Zhuolin Qu

Thesis (M. Phil.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.

GaAs-based nitride is found to be sensitive to growth conditions and ex-situ annealing processes. The critical thickness is almost one order thicker than the theoretical prediction by force balance model. The growth process could be sped up by the nitrogen incorporation itself, while the nitrogen incorporation could be affected by Beryllium doping. The incorporated nitrogen atoms partly occupy substitutional sites for Arsenic. Some nitrogen atoms are at interstitial sites. Annealing could drastically increase the optical quality of GaAs-based nitrides. As an end of this paper, some preliminary results of InGaAsN/GaAsN/AlGaAs laser diodes are also presented. / Singapore-MIT Alliance (SMA)

GaAs-based InGaAsN/GaAs quantum well is found to be very sensitive to growth conditions and ex-situ annealing processes. Annealing could drastically increase the optical quality of GaAs-based InGaAsN/GaAs quantum well. As an end of this paper, some results on InGaAsN/GaAsN/AlGaAs laser diodes are also presented. / Singapore-MIT Alliance (SMA)

In this article, we report an attempt of extending the InGaAsN materials towards 1.3µm and 1.55µm wavelength. All these InGaAsN samples are grown in a plasma-assisted solid-source molecular-beam epitaxy (SS-MBE) system. Our experiments revealed that the nitrides could be grown with both direct nitrogen beam and dispersive nitrogen. The nitrogen incorporation rate could be reduced by the presence of indium flux. The interaction between nitrogen and indium might lead to 3D growth mode and growth dynamics. It is proved that the increasing growth rate reduces the nitrogen incorporation efficiency. The data for nitrogen sticking coefficient are somewhat contradictive. The growth with dispersive nitrogen source causes the improvement of material quality. Fixed indium flux is a better way for the wavelength control. Also, we report some growth optimization work for better PL property and the annealing effect on the samples. Literature is sometimes reviewed for comparison. / Singapore-MIT Alliance (SMA)

Thesis (Ph. D.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

The effect of silicon doping on the strain in c-plane InN films grown on c-plane GaN by plasma-assisted molecular beam epitaxy is investigated. Strain is measured by x-ray reciprocal space mapping and Raman spectroscopy. The silicon doping concentration of our sample is about 1018 cm-3 by Hall measurement. Relation between the strain and the silicon concentration is obtained. To understand the increase in tensile stress caused by Si doping is discussed in terms of a crystallite coalescence model.

We study the properties of m-plane GaN structure on LiAlO 2 substrate grown by plasma-assisted molecular-beam epitaxy (PAMBE). Lattice parameters of LiAlO 2 are close to GaN, the interface between LiAlO 2 and GaN showed a good lattice matching. Low lattice mismatch can reduce the defect generation, improve crystal quality. However, lattice mismatch still exist, more or less density of defect still can be observed. The density of defect was reduced in the sample at high temperature. In this study, we investigate GaN on LiAlO 2 by scanning electron microscope (SEM), atomic force microscope (AFM), photoluminescence (PL) and X-ray diffraction (XRD) for different growth temperatures.

¡@In this thesis, we have studied the growth of M-plane GaN thin film on LiGaO2 (100) substrate by Plasma-Assisted Molecular Beam Epitaxy. We found that the growth of GaN thin films on as-received LiGaO2 substrates is poly-crystalline by analysis of X-ray diffraction, and these of GaN thin films were peeled off after thin film process. Using atomic force microscopy (AFM) to scan the surface of as-received LiGaO2 substrate, we found that many particles which are Ga2O3 existed on the surface of as-received LiGaO2. The annealing ambient for LiGaO2 substrates in vacuum and air ambient has been studied in order to improve the surface of LiGaO2. The scanning results of AFM shows that the crystal quality and stress of M-plane GaN grown on LiGaO2 (100) substrate pre-annealed in air ambient is significantly improved. We conclude that the reason of GaN peeling off from LiGaO2 substrate is attributed to stress between GaN/ LiGaO2. The measurement of polarization-dependent PL shows that the luminescence intensity of growing sample increases and reaches a maximum at £p = 90¢X (E¡æc), which indicates the growing samples is M-plane GaN as well. The microstructure of growing samples was characterized by transmission electron microscopy. We found that the formation of stacking fault in GaN is attributed to the growth of GaN on cubic-Ga2O3 nano-particles. The formation of Ga2O3 nano-particles can be suppressed by pre-annealing LiGaO2 substrate in air. It revealed that the thermal annealing LiGaO2 substrate in air ambient can improve the surface of LiGaO2 substrate effectively, and then one can grow a high quality M-plane GaN thin film on the LiGaO2 substrate.