Finite difference

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

A finite volume numerical approach for coastal ocean circulation studies: Comparisons with finite difference models

Abstract: [1] An unstructured grid, finite volume, three-dimensional (3-D) primitive equation coastal ocean model (FVCOM) has been developed for the study of coastal ocean and estuarine circulation by Chen et al. (2003a). The finite volume method used in this model combines the advantage of finite element methods for geometric flexibility and finite difference methods for simple discrete computation. Currents, temperature, and salinity are computed using an integral form of the equations, which provides a better representation of the conservative laws for mass, momentum, and heat. Detailed comparisons are presented here of FVCOM simulations with analytical solutions and numerical simulations made with two popular finite difference models (the Princeton Ocean Model and Estuarine and Coastal Ocean Model (ECOM-si)) for the following idealized cases: wind-induced long-surface gravity waves in a circular lake, tidal resonance in rectangular and sector channels, freshwater discharge onto the continental shelf with curved and straight coastlines, and the thermal bottom boundary layer over the slope with steep bottom topography. With a better fit to the curvature of the coastline using unstructured nonoverlapping triangle grid cells, FVCOM provides improved numerical accuracy and correctly captures the physics of tide-, wind-, and buoyancy-induced waves and flows in the coastal ocean. This model is suitable for applications to estuaries, continental shelves, and regional basins that feature complex coastlines and bathymetry.

Higher Order Triangular Finite Elements with Mass Lumping for the Wave Equation

Abstract: In this article, we construct new higher order finite element spaces for the approximation of the two-dimensional (2D) wave equation. These elements lead to explicit methods after time discretization through the use of appropriate quadrature formulas which permit mass lumping. These formulas are constructed explicitly. Error estimates are provided for the corresponding semidiscrete problem. Finally, higher order finite difference time discretizations are proposed and various numerical results are shown.

Multiscale and Stabilized Methods

Abstract: : This article presents an introduction to multiscale and stabilized methods, which represent unified approaches to modeling and numerical solution of fluid dynamic phenomena. Finite element applications are emphasized but the ideas are general and apply to other numerical methods as well. (They have been used in the development of finite difference, finite volume, and spectral methods, in addition to finite element methods.) The analytical ideas are first illustrated for time-harmonic wave-propagation problems in unbounded fluid domains governed by the Helmholtz equation. This leads to the well-known Dirichlet-to-Neumann formulation. A general treatment of the variational multiscale method in the context of an abstract Dirichlet problem is then presented which is applicable to advective-diffusive processes and other processes of physical interest. It is shown how the exact theory represents a paradigm for subgrid-scale models and posteriori error estimation. Hierarchical p-methods and bubble function methods are examined in order to understand and, ultimately, approximate the "fine-scale Green's function" which appears in the theory. Relationships among so-called residual-free bubbles, element Green's functions, and stabilized methods are exhibited. These ideas are then generalized to a class of non-symmetric, linear evolution operators formulated in space-time. The variational multiscale method also provides guidelines and inspiration for the development of stabilized methods which have attracted considerable interest and have been extensively utilized in engineering and the physical sciences. An overview of stabilized methods for advective-diffusive equations is presented. A variational multiscale treatment of incompressible viscous flows, including turbulence is also described. This represents an alternative formulation of Large Eddy Simulation which provides simplified theoretical framework of LES with potential for improved modeling.

Multiple scattering as a diffusion process

Abstract: Multiple scattering in participating media is generally a complex phenomenon. In the limit of an optically thick medium, i.e., when the mean free path of each photon is much smaller than the medium size, the effects of multiple scattering can be approximated by a diffusion process. We introduce this approximation from the radiative transfer literature to the computer graphics community and propose several numerical methods for its solution. We implemented both a multi-grid finite differences scheme and a finite-element blob method.