A level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of a two-phase flow where the interface can merge/break and the flow can have a high Reynolds number. A distance function formulation of the level set method enables us to compute flows with large density ratios (1000/1) and flows that are surface tension driven, with no emotional involvement. Recent work has improved the accuracy of the distance function formulation and the accuracy of the advection scheme. We compute flows involving air bubbles and water drops, among others. We validate our code against experiments and theory.

New numerical methods are derived for calculation of high frequency asymptotic expansion of the scalar wave equation. The nonlinear partial differential equations defining the terms in the expansion are approximated directly rather than via ray tracing. High resolution numerical algorithms are used to handle discontinuities and new devices are introduced to represent the multivalued character of the solution.

We present a new numerical method for solving the Boltzmann-Poisson system which describes charge transport in semiconductor devices. The Boltzmann equation is reduced from three dimensions in velocity space to two by taking the electric field parallel to the z axis, which implies invariance of the probability density function under rotation around the z axis. We develop a finite difference discretization of the Boltzmann equation in one spatial dimension and two-dimensional velocity space, coupled to the Poisson equation. The system of equations obtained by taking the first five moments of the Boltzmann equation coupled to the Poisson equation is known as the hydrodynamic model in semiconductor modeling. A comparison of the numerical results from our method and the hydrodynamic model is given. Also a numerical investigation is done with respect to the heat conduction, viscosity, and momentum relaxation terms in the hydrodynamic model.

A constrained optimization type of numerical algorithm for removing noise from images is presented. The total variation of the image is minimized subject to constraints involving the statistics of the noise. The constraints are imposed using Lanrange multipliers. The solution is obtained using the gradient-projection method. This amounts to solving a time dependent partial differential equation on a manifold determined by the constraints. As t → ∞ the solution converges to a steady state which is the denoised image. The numerical algorithm is simple and relatively fast. The results appear to be state-of-the-art for very noisy images. The method is noninvasive, yielding sharp edges in the image. The technique could be interpreted as a first step of moving each level set of the image normal to itself with velocity equal to the curvature of the level set divided by the magnitude of the gradient of the image, and a second step which projects the image back onto the constraint set.

Simulation results for the hydrodynamic model are presented for an n/sup +/-n-n/sup +/ diode by use of shock-capturing numerical algorithms applied to the transient model with subsequent passage to the steady state. The numerical method is first order in time, but of high spatial order in regions of smoothness. Implementation typically requires a few thousand time steps. These algorithms, termed essentially nonoscillatory, have been successfully applied in other contexts to model the flow in gas dynamics, magnetohydrodynamics, and other physical situations involving the conservation laws of fluid mechanics. The presented semiconductor simulations reveal temporal and spatial velocity overshot, as well as overshoot relative to an electric field induced by the Poisson equation. Shocks are observed in the transient simulations for certain low-temperature parameter regimes.< >

Finite difference methods are developed to solve the nonlinear PDEs approximating somesolutions of the Helmholtz equation in high frequency regime. Numerical methods are developedfor solving the geometrical optics approximation, classical asymptotic expansion,and a new perturbed geometrical optics system. Difficulties with the geometrical opticsand related approximations associated with the multivaluedness of the phase function arediscussed. The geometrical optics fails to represent...