Topic
Finite difference
About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.
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01 Jan 1999
TL;DR: In this article, the authors present analytical models for linear and nonlinear time series, as well as an analytical model for cellular automata and lattice gases, and a model for partial differential equations.
Abstract: Preface 1. Introduction Part I. Analytical Models: 2. Ordinary differential and difference equations 3. Partial differential equations 4. Variational principles 5. Random systems Part II. Numerical Models: 6. Finite differences: ordinary difference equations 7. Finite differences: partial differential equations 8. Finite elements 9. Cellular automata and lattice gases Part III. Observational Models: 10. Function fitting 11. Transforms 12. Architectures 13. Optimization and search 14. Clustering and density estimation 15. Filtering and state estimation 16. Linear and nonlinear time series Appendix 1. Graphical and mathematical software Appendix 2. Network programming Appendix 3. Benchmarking Appendix 4. Problem solutions Bibliography.
460 citations
01 Nov 2002
TL;DR: An accurate and efficient numerical method to solve the coupled Cahn-Hilliard/Navier-Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity, and solves the Navier- Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank-Nicolson-based projection methods.
Abstract: Phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near-critical interfacial phenomena, phase separation under shear, and microstructure evolution during solidification. However, because interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations require resolution of very thin layers to capture the physics of the problems studied. This demands robust numerical methods that can efficiently achieve high resolution and accuracy, especially in three dimensions. We present here an accurate and efficient numerical method to solve the coupled Cahn-Hilliard/Navier-Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity. The numerical method is a time-split scheme that combines a novel semi-implicit discretization for the convective Cahn-Hilliard equation with an innovative application of high-resolution schemes employed for direct numerical simulations of turbulence. This new semi-implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the Cahn-Hilliard equation at the same cost as that of an explicit scheme. It is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity. Moreover, we solve the Navier-Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank-Nicolson-based projection methods. For channel geometries, the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction. The capabilities of the method are demonstrated with several examples including phase separation with, and without, shear in two and three dimensions. The method effectively resolves interfacial layers of as few as three mesh points. The numerical examples show agreement with analytical solutions and scaling laws, where available, and the 3D simulations, in the presence of shear, reveal rich and complex structures, including strings.
456 citations
TL;DR: In this article, a 2D thermal-mechanical code incorporating both a Lagrangian marker-in-cell method and conservative finite-difference (FD) schemes was designed for the accurate portrayal of very fine thermal structures.
455 citations
01 Jan 1974
TL;DR: In this paper, the authors define finite element and finite difference methods for hyperbolic partial differential equations and show that the resulting procedures are automatically stable and there is extreme flexibility in choosing the basic functions, therefore, in very complicated domains or for problems with complicated interfaces, the method is the only feasible one.
Abstract: Publisher Summary
This chapter defines finite element and finite difference methods for hyperbolic partial differential equations. The advantage of the finite element method is that the resulting procedures are automatically stable and there is extreme flexibility in choosing the basic functions. Therefore, in very complicated domains or for problems with complicated interfaces, the method is the only feasible one. For hyperbolic partial differential equations it is essential to control the dispersion, dissipation, and the propagation of discontinuities. This is easily done by using suitable difference approximations. The main disadvantage of finite difference methods is that it may be difficult to handle boundaries properly.
454 citations
TL;DR: A new mode solver is described which uses Yee's 2-D mesh and an index averaging technique to provide a full-vectorial finite-difference analysis of microstructured optical fibers.
Abstract: In this paper we present a full-vectorial finite-difference analysis of microstructured optical fibers. A new mode solver is described which uses Yee's 2-D mesh and an index averaging technique. The modal characteristics are calculated for both conventional optical fibers and microstructured optical fibers. Comparison with previous finite difference mode solvers and other numerical methods is made and excellent agreement is achieved.
450 citations