We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations, including Hamilton-Jacobi equations, quasi-linear hyperbolic equations, and conservative transport equations with multi-valued transport speeds. The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space. We discuss the essential ideas behind the techniques, the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the computation of the semiclassical limit for Schrödinger equations and the high frequency geometrical optics limits of linear wave equations. 

An adaptive moving mesh method is developed for the numerical solution of two-dimensional phase change problems modelled by the phase-field equations. The numerical algorithm is relatively simple and is shown to be more efficient than fixed grid methods. The phase-field equations are discretized by a Galerkin finite element method. An adaptivity criterion is used that ensures that the mesh spacing at the phase front scales with the diffuse interface thickness.

This paper is concerned with the numerical simulation of multiphase, multicomponent flow in porous media. The model equations are based on compositional flow with mass interchange between phases. The compositional model consists of Darcy's law for volumetric flow velocities, mass conservation for hydrocarbon components, thermodynamic equilibrium for mass interchange between phases, and an equation of state for saturations. High-accurate finite volume methods on unstructured grids are used to discretize the model governing equations. Special emphasis is placed on studying the influence of gravitational effects on the overall displacement dynamics. In particular, free and forced convections, diffusions, and dispersions are studied in separate and combined cases, and their interplays are intensively analyzed for gravitational instabilities. Extensive numerical experiments are presented to validate the numerical study under consideration.

In this work, noise removal in digital images is investigated. The importance of this problem lies in the fact that removal of noise is a necessary pre-processing step for other image processing tasks such as edge detection, image segmentation, image compression, classification problems, image registration etc. A number of different approaches have been proposed in the literature. In this work, a non-linear PDE-based algorithm is developed based on the ideas proposed by Lysaker, Osher and Tai [IEEE Trans. Image Process., 13 (2004), 1345-1357] . This algorithm consists of two steps: flow field smoothing of the normal vectors, followed by image reconstruction. We propose a finite-difference based additive operator-splitting method that allows for much larger time-steps. This results in an efficient method for noise-removal that is shown to have good visual results. The energy is studied as an objective measure of the algorithm performance.

The Doi-Hess equation that describes the evolution of an orientational distribution function is capable of predicting several rheological features of nematic polymers. Since the orientational distribution function becomes sharply peaked as potential intensity increases, powerful numerical methods become necessary in the relevant numerical simulations. In this paper, a numerical scheme based on the moving grid techniques will be designed to solve the orientational distribution functions with high potential intensities. Numerical experiments are carried out to demonstrate the effectiveness and robustness of the proposed scheme. 

An augmented method is proposed for solving stationary incompressible Stokes equations with a Dirichlet boundary condition along parts of the boundary. In this approach, the normal derivative of the pressure along the parts of the boundary is introduced as an additional variable and it is solved by the GMRES iterative method. The dimension of the augmented variable in discretization is the number of grid points along the boundary which is O(N). Each GMRES iteration (or one matrix-vector multiplication) requires three fast Poisson solvers for the pressure and the velocity. In our numerical experiments, only a few iterations are needed. We have also combined the augmented approach for Stokes equations involving interfaces, discontinuities, and singularities. 

Total and state-selective charge transfer, ionization and stripping cross sections due to the collision of Si²⁺ ion with atomic hydrogen are investigated using the classical-trajectory Monte-Carlo (CTMC) method in the collision energy from 1 keV/amu to 10 MeV/amu. Total electron capture rate coefficient is obtained in the temperature range from 10⁵ K^o to 10⁸ K^o. Comparison with the data available shows that our CTMC results are reliable. The behaviors for these cross sections varying with the projectile energy are analyzed. A classical physical picture is presented to explain the reason behind the behaviors.

In this work, the modified Ghost Fluid Method is further developed to apply to compressible fluid coupled to deformable structure, where the pressure in the structure or flow can vary from an initial extremely high magnitude (such that the solid medium can be under plastic compression) to a subsequently very low quantity (so that cavitation can occur in the fluid). New techniques are also developed in the definition of the ghost fluid status when the structure is under plastic deformation or when the flow is under cavitation next to the structure. Numerical results show that the improved MGFM for treatment of the fluid-deformable structure coupling works efficiently for all pressure ranges and is capable of simulating cavitation evolution and cavitation reloading in conjunction with the employment of the isentropic one-fluid cavitation model.

The transverse construction of optical modes in the wiggler is calculated and the numerical simulations of the free-electron laser hole-coupled resonator are carried out. These 3D-simulations include optical amplitude distributions, modes evolvement of optical field and the influence of hole-radius on the distributing of modes. The numerical simulations confirm that the fraction of the even-order modes increase in the start-up stage and then decreases in the exponential gain stage. Moreover, it is found that the fundamental mode is dominant in the saturate stage. Based on this observation, we estimate the optical output coupling by using the the fundamental mode. It is found that the numerical results and the first-order estimate are in good agreement for a range of the hole size.

We develop an efficient one-dimensional moving mesh algorithm for solving partial differential equations. The main contribution of this paper is to design an effective interpolation scheme based on L²-projection for the moving mesh method. The proposed method preserves not only the mass-conservation but also the first order momentum of the underlying numerical solution at each mesh redistribution step. Numerical examples are presented to demonstrate the effectiveness of the new interpolation technique.