Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

Rigorous three-dimensional time-domain finite-difference electromagnetic simulation

Abstract: This thesis describes the latest embodiment of a three-dimensional electromagnetic simulation program called TEMPEST which is implemented on the connection machines CM-2 and CM-5, and is used to predict and study technology trade-offs of interest in photolithography. Highlights of the new algorithm include generalization to three-dimensional calculation, analysis of dispersive materials, an efficient absorbing boundary condition, oblique incidence, and image synthesis based on Hopkins' formulation. The finite propagation speed of electromagnetic waves makes the time-domain finite-difference approach a natural choice for implementation on parallel computer architectures. This thesis addresses algorithmic issues including the accuracy and stability of the numerical scheme, and numerical boundary conditions. The conventional time-domain finite-difference scheme is second order accurate and requires 15 simulation nodes per wavelength to achieve a 2% accuracy. Stability of the scheme depends on the ratio between the spatial and temporal discretizations. Analogous to previous work in plasma physics, instability of the algorithm due to highly dispersive materials has been alleviated by calculating explicitly the time-domain convolution relation between the electric field and the electric displacement. A novel boundary condition derived from the harmonic nature of electromagnetic waves is used to bound the simulation domain with minimal artificial reflection. Implementation of a software package which caters to user convenience in data processing and remote use of the connection machine is also included. A link between TEMPEST and the optical image simulation program SPLAT allows the study of the interaction among mask topography effects, partial coherence effects, and lens aberrations. Validation of TEMPEST via experimental comparison and the usefulness of the program in predicting and assessing complex technological issues in photolithography are presented. In particular, TEMPEST is used to predict important effects such as glass edge scattering in phase-shifting masks and resonance in dielectric ridges. These predictions have been subsequently confirmed experimentally. Transmission loss and polarization effects in small contact holes are characterized as a function of the feature size. TEMPEST is also shown to be well-suited for analyzing three-dimensional effects of reflection from underlying topography during photoresist exposure which can cause variations in the critical dimensions of the features being formed.

A general circulation model of the atmosphere suitable for long period integrations

Abstract: The design of a general circulation model using the primitive equations in spherical form is described, including a statement of the finite difference forms used to integrate the system and explanations of the motives for unusual aspects of the finite difference scheme. The model incorporates the hydrological cycle, topography, a simple scheme for the radiative exchanges and arrangements for the simulation of deep free convection (sub grid-scale) and for the representation of exchanges of momentum, sensible and latent heat with the underlying surface. An experiment performed with the model forms the subject of a separate paper.

On the maximum principle preserving schemes for the generalized Allen–Cahn equation

Abstract: This paper is concerned with the generalized Allen-Cahn equation with a nonlinear mobility that may be degenerate, which also includes an advection term appearing in many phase- field models for multi-component fluid flows. A class of maximum principle preserving schemes will be studied for the generalized Allen-Cahn equation, with either the commonly used polynomial free energy or the logarithmic free energy, and with a nonlinear degenerate mobility. For time discretization, the standard semi-implicit scheme as well as the stabilized semi-implicit scheme will be adopted, while for space discretization, the central finite difference is used for approximating the diffusion term and the upwind scheme is employed for the advection term. We establish the maximum principle for both semi-discrete (in time) and fully discretized schemes. We also provide an error estimate by using the established maximum principle which plays a key role in the analysis. Several numerical experiments are carried out to verify our theoretical results.

The Henry Problem for Saltwater Intrusion

Abstract: Previous solutions of the idealized saltwater intrusion problem known as the Henry problem are discussed, and possible reasons for the observed discrepancies between them are given. High-accuracy finite difference techniques are used to solve the nondimensionalized equations governing the problem, and a fine grid is used so that the solutions obtained contain only very small truncation errors. Such errors are investigated by means of grid refinement. Comparison of past results with the present solutions indicate, first, the presence of significant inaccuracy in certain earlier results and, second, the effects of numerical dispersion in other previous solutions calculated using relatively few grid points.