Showing papers on "Finite difference published in 2008"
TL;DR: This paper develops an implicit unconditionally stable numerical method to solve the one-dimensional linear time fractional diffusion equation, formulated with Caputo's fractional derivative, on a finite slab.
Abstract: Time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical Brownian motion model In this paper we develop an implicit unconditionally stable numerical method to solve the one-dimensional linear time fractional diffusion equation, formulated with Caputo's fractional derivative, on a finite slab Several numerical examples of interest are also included
393 citations
TL;DR: In this paper, a new full-vector finite difference discretization based on transverse magnetic field components was proposed for calculating the electromagnetic modes of optical waveguides with transverse, non-diagonal anisotropy.
Abstract: We describe a new full-vector finite difference discretization, based upon the transverse magnetic field components, for calculating the electromagnetic modes of optical waveguides with transverse, nondiagonal anisotropy. Unlike earlier finite difference approaches, our method allows for the material axes to be arbitrarily oriented, as long as one of the principal axes coincides with the direction of propagation. We demonstrate the capabilities of the method by computing the circularly-polarized modes of a magnetooptical waveguide and the modes of an off-axis poled anisotropic polymer waveguide.
371 citations
TL;DR: In this article, a benchmark comparison between various numerical codes (Eulerian and Lagrangian, Finite Element and Finite Difference, with and without markers) as well as a laboratory experiment is presented.
365 citations
TL;DR: This work presents a critical review of the main conventional methods for multiphase flow in fractured media including the finite difference, finite volume, and finite element methods, that are coupled with the discrete-fracture model and introduces a new approach that is free from the limitations of the conventional methods.
266 citations
TL;DR: This paper uses the first few eigenfunctions of the backward Fokker–Planck diffusion operator as a coarse-grained low dimensional representation for the long-term evolution of a stochastic system and shows that they are optimal under a certain mean squared error criterion.
Abstract: The concise representation of complex high dimensional stochastic systems via a few reduced coordinates is an important problem in computational physics, chemistry, and biology. In this paper we use the first few eigenfunctions of the backward Fokker–Planck diffusion operator as a coarse-grained low dimensional representation for the long-term evolution of a stochastic system and show that they are optimal under a certain mean squared error criterion. We denote the mapping from physical space to these eigenfunctions as the diffusion map. While in high dimensional systems these eigenfunctions are difficult to compute numerically by conventional methods such as finite differences or finite elements, we describe a simple computational data-driven method to approximate them from a large set of simulated data. Our method is based on defining an appropriately weighted graph on the set of simulated data and computing the first few eigenvectors and eigenvalues of the corresponding random walk matrix on this graph...
265 citations
TL;DR: This work presents spectral element and discontinuous Galerkin solutions of the Euler and compressible Navier-Stokes equations for stratified fluid flow which are of importance in nonhydrostatic mesoscale atmospheric modeling and recommends the DG method due to its conservation properties.
243 citations
218 citations
TL;DR: In this article, the Navier-Stokes equations were solved with a finite difference numerical method, which proved to be highly stable even at very high Reynolds numbers, and compared with experimental and numerical results found in the literature.
216 citations
TL;DR: Extensions of finite-difference time domain (FDTD) and finite-element time-domain (FETD) algorithms are reviewed for solving transient Maxwell equations in complex media in this article.
Abstract: Extensions of finite-difference time-domain (FDTD) and finite-element time-domain (FETD) algorithms are reviewed for solving transient Maxwell equations in complex media. Also provided are a few representative examples to illustrate the modeling capabilities of FDTD and FETD for complex media. The term complex media refers here to media with dispersive, (bi)anisotropic, inhomogeneous, and/or nonlinear properties present in the constitutive tensors.
210 citations
TL;DR: It is proved that nonconservative schemes generate, at the level of the limiting hyperbolic system, an convergence error source-term which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure.
203 citations
TL;DR: A stable wall boundary procedure is derived for the discretized compressible Navier-Stokes equations using high-order accurate finite difference summation-by-parts operators and it is proved linear stability for the scheme including the wall boundary conditions.
TL;DR: In this article, the in-plane properties for hexagonal and reentrant (auxetic) lattices are investigated through the analysis of partial differential equations associated with their homogenized continuum models, and the estimation of the mechanical properties is carried out through a comparison between the derived differential equations and appropriate elasticity models.
TL;DR: In this article, a convergent monotone finite difference scheme for the Monge-Ampere equation, Pucci's Maximal and Minimal Equations, and the convex envelope is presented.
Abstract: Certain fully nonlinear elliptic Partial Differential Equations can be written as functions of the eigenvalues of the Hessian. These include: the Monge-Ampere equation, Pucci’s Maximal and Minimal equations, and the equation for the convex envelope. In this article we build convergent monotone finite difference schemes for the aforementioned equations. Numerical results are presented.
TL;DR: A formulation of local high-order ABCs recently proposed by Hagstrom and Warburton and based on a modification of the Higdon ABCs, is further developed and extended in a number of ways.
Book•
04 Sep 2008
TL;DR: Researchers who simulate processes that are modeled by evolutionary differential equations will find material on the principles underlying the appropriate method to use and the pitfalls that accompany each method useful.
Abstract: Methods for the numerical simulation of dynamic mathematical models have been the focus of intensive research for well over 60 years, and the demand for better and more efficient methods has grown as the range of applications has increased. Mathematical models involving evolutionary partial differential equations (PDEs) as well as ordinary differential equations (ODEs) arise in diverse applications such as uid ow, image processing and computer vision, physics-based animation, mechanical systems, relativity, earth sciences, and mathematical nance. This textbook develops, analyzes, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both PDEs and ODEs are discussed from a unified viewpoint. The author emphasizes finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and nonsmooth solutions for hyperbolic PDEs, parabolic-type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is included as well. The author bridges theory and practice by developing algorithms, concepts, and analysis from basic principles while discussing efficiency and performance issues and demonstrating methods through examples and case studies from numerous application areas. Audience: This textbook is suitable for researchers and graduate students from a variety of fields including computer science, applied mathematics, physics, earth and ocean sciences, and various engineering disciplines. Gradute students at the beginning or advanced level (depending on the discipline) and researchers in a variety of fields in science and engineering will find this book useful. Researchers who simulate processes that are modeled by evolutionary differential equations will find material on the principles underlying the appropriate method to use and the pitfalls that accompany each method. Contents: Preface; 1 Introduction; 2 Methods and Concepts for ODEs; 3 Finite Difference and Finite Volume Methods; 4 Stability for Constant Coefficient Problems; 5 Variable Coefficient and Nonlinear Problems; 6 Hamiltonian Systems and Long Time Integration; 7 Dispersion and Dissipation; 8 More on Handling Boundary Conditions; 9 Several Space Variables and Splitting Methods; 10 Discontinuities and Almost Discontinuities; 11 Additional Topics; Bibliography; Index.
TL;DR: In this article, a code for solving the coupled Einstein-hydrodynamics equations to evolve relativistic, self-gravitating fluids is presented, which accurately evolves equilibrium stars and accretion flows.
Abstract: We present a code for solving the coupled Einstein-hydrodynamics equations to evolve relativistic, self-gravitating fluids. The Einstein field equations are solved in generalized harmonic coordinates on one grid using pseudospectral methods, while the fluids are evolved on another grid using shock-capturing finite difference or finite volume techniques. We show that the code accurately evolves equilibrium stars and accretion flows. Then we simulate an equal-mass nonspinning black hole-neutron star binary, evolving through the final four orbits of inspiral, through the merger, to the final stationary black hole. The gravitational waveform can be reliably extracted from the simulation.
TL;DR: In this article, a model order reduction method is developed and applied to 1D diffusion systems with negative real eigenvalues, and residues with similar eigen values are grouped together to reduce the model order.
Abstract: A model order reduction method is developed and applied to 1D diffusion systems with negative real eigenvalues. Spatially distributed residues are found either analytically (from a transcendental transfer function) or numerically (from a finite element or finite difference state space model), and residues with similar eigenvalues are grouped together to reduce the model order. Two examples are presented from a model of a lithium ion electrochemical cell. Reduced order grouped models are compared to full order models and models of the same order in which optimal eigenvalues and residues are found numerically. The grouped models give near-optimal performance with roughly 1/20 the computation time of the full order models and require 1000―5000 times less CPU time for numerical identification compared to the optimization procedure.
TL;DR: In this paper, an improved immersed boundary method for simulating incompressible viscous flow around an arbitrarily moving body on a fixed computational grid was presented, which combined the feedback foreing scheme of the virtual boundary method with Peskin's regularized delta function approach.
Abstract: We present an improved immersed boundary method for simulating incompressible viscous flow around an arbitrarily moving body on a fixed computational grid. To achieve a large CFL number and to transfer quantities between Eulerian and Lagrangian domains effectively, we combined the feedback foreing scheme of the virtual boundary method with Peskin’s regularized delta function approach. Stability analysis of the proposed method was carried out for various types of regularized delta function. The stability regime of the 4-point regularized delta function was much wider than that of the 2-point delta function. An optimum regime of the feedback forcing is suggested on the basis of the analysis of stability limits and feedback forcing gains. The proposed method was implemented in a finite difference and fractional step context. The proposed method was tested on several flow problems and the findings were in excellent agreement with previous numerical and experimental results.
TL;DR: In this paper, a new method for solving the time-domain integral equations of electromagnetic scattering from conductors is introduced, called finite difference delay modeling, which appears to be completely stable and accurate when applied to arbitrary structures.
Abstract: A new method for solving the time-domain integral equations of electromagnetic scattering from conductors is introduced. This method, called finite difference delay modeling, appears to be completely stable and accurate when applied to arbitrary structures. The temporal discretization used is based on finite differences. Specifically, based on a mapping from the Laplace domain to the z-transform domain, first- and second-order unconditionally stable methods are derived. Spatial convergence is achieved using the higher-order divergence-conforming vector bases of Graglia et al. Low frequency instability problems are avoided with the loop-tree decomposition approach. Numerical results will illustrate the accuracy and stability of the technique.
TL;DR: This work presents an approach, based on work by Miehe, for an efficient numerical approximation of the tangent moduli that can be easily implemented within commercial FE codes and will facilitate the incorporation of novel hyperelastic material models for a soft tissue behavior into commercial FE software.
Abstract: Finite element (FE) implementations of nearly incompressible material models often employ decoupled numerical treatments of the dilatational and deviatoric parts of the deformation gradient. This treatment allows the dilatational stiffness to be handled separately to alleviate ill conditioning of the tangent stiffness matrix. However, this can lead to complex formulations of the material tangent moduli that can be difficult to implement or may require custom FE codes, thus limiting their general use. Here we present an approach, based on work by Miehe (Miehe, 1996, "Numerical Computation of Algorithmic (Consistent) Tangent Moduli in Large Strain Computational Inelasticity," Comput. Methods Appl. Mech. Eng., 134, pp. 223-240), for an efficient numerical approximation of the tangent moduli that can be easily implemented within commercial FE codes. By perturbing the deformation gradient, the material tangent moduli from the Jaumann rate of the Kirchhoff stress are accurately approximated by a forward difference of the associated Kirchhoff stresses. The merit of this approach is that it produces a concise mathematical formulation that is not dependent on any particular material model. Consequently, once the approximation method is coded in a subroutine, it can be used for other hyperelastic material models with no modification. The implementation and accuracy of this approach is first demonstrated with a simple neo-Hookean material. Subsequently, a fiber-reinforced structural model is applied to analyze the pressure-diameter curve during blood vessel inflation. Implementation of this approach will facilitate the incorporation of novel hyperelastic material models for a soft tissue behavior into commercial FE software.
TL;DR: The numerical solution of the time fractional inverse heat conduction problem (TFIHCP) on a finite slab is investigated in the presence of measured (noisy) data when the time fractionsal derivative is interpreted in the sense of Caputo.
Abstract: The numerical solution of the time fractional inverse heat conduction problem (TFIHCP) on a finite slab is investigated in the presence of measured (noisy) data when the time fractional derivative is interpreted in the sense of Caputo. A finite difference space marching scheme with adaptive regularization, using mollification techniques, is introduced. Error estimates are derived for the numerical solution of the mollified problem and several numerical examples of interest are provided.
TL;DR: In this article, a continuous model for dynamics of large-diameter sagged cables, taking bending stiffness and sag extensibility into account, is presented. But the model is not suitable for large diameter sagged cable.
TL;DR: In this article, the authors compared the accuracies of various finite difference (FD), finite element (FE) and spectral methods (SDF) for the Stokes equations for creeping, highly viscous flows.
TL;DR: A time stable discretization is derived for the second-order wave equation with discontinuous coefficients using narrow-diagonal summation by parts operators and patched to its neighbors by using a penalty method, leading to fully explicit time integration.
TL;DR: In this article, the authors obtained an algebraic rate of convergence for monotone and consistent finite difference approximations to Lipschitz-continuous viscosity solutions of uniformly elliptic partial differential equations.
Abstract: We obtain an algebraic rate of convergence for monotone and consistent finite difference approximations to Lipschitz-continuous viscosity solutions of uniformly elliptic partial differential equations. © 2007 Wiley Periodicals, Inc.
TL;DR: In this paper, the singular forces are computed implicitly by solving a small system of equations at each time step, which is derived from a second order projection method and then distributed to nearby Cartesian grid points using the immersed boundary method.
TL;DR: A numerical scheme for solving type boundary value problems, which works nicely in both the cases, i.e., when delay argument is bigger one as well as smaller one, is presented.
TL;DR: In this article, numerical solutions of the Benjamin-Bona-Mahony-Burgers equation in one space dimension are considered using Crank-Nicolson-type finite difference method.
Abstract: Numerical solutions of the Benjamin-Bona-Mahony-Burgers equation in one space dimension are considered using Crank-Nicolson-type finite difference method. Existence of solutions is shown by using the Brower's fixed point theorem. The stability and uniqueness of the corresponding methods are proved by the means of the discrete energy method. The convergence in L∞-norm of the difference solution is obtained. A conservative difference scheme is presented for the Benjamin-Bona-Mahony equation. Some numerical experiments have been conducted in order to validate the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007
TL;DR: First-order elliptic Conditional Moment Closure (CMC), coupled with a computational fluid dynamics (CFD) solver, has been employed to simulate combustion in a direct-injection heavy-duty diesel engine.
Abstract: First-order elliptic Conditional Moment Closure (CMC), coupled with a computational fluid dynamics (CFD) solver, has been employed to simulate combustion in a direct-injection heavy-duty diesel engine. The three-dimensional structured finite difference CMC grid has been interfaced to an unstructured finite volume CFD mesh typical of engine modelling. The implementation of a moving CMC grid to reflect the changes in the domain due to the compression and expansion phases has been achieved using an algorithm for the cell addition/removal and modelling the additional convection term due to the CMC cell movement. Special care has been taken for the boundary conditions and the wall heat transfer. An operator splitting formulation has been used to integrate the CMC equations efficiently. A CMC domain reduction of the three-dimensional problem to two- and zero-dimensions through appropriate volume integration of the CMC equation has been explored in terms of accuracy and computational time. Additional considerati...
TL;DR: In this article, a method for accurately describing arbitrary-shaped free boundaries in single-grid finite-difference schemes for elastodynamics, in a time-domain velocity-stress framework, is proposed.
Abstract: A method is proposed for accurately describing arbitrary-shaped free boundaries in single-grid finite-difference schemes for elastodynamics, in a time-domain velocity-stress framework. The basic idea is as follows: fictitious values of the solution are built in vacuum, and injected into the numerical integration scheme near boundaries. The most original feature of this method is the way in which these fictitious values are calculated. They are based on boundary conditions and compatibility conditions satisfied by the successive spatial derivatives of the solution, up to a given order that depends on the spatial accuracy of the integration scheme adopted. Since the work is mostly done during the preprocessing step, the extra computational cost is negligible. Stress-free conditions can be designed at any arbitrary order without any numerical instability, as numerically checked. Using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths yields highly accurate results. With 5 grid nodes per minimal S-wavelength, the solution is less accurate but still acceptable. A subcell resolution of the boundary inside the Cartesian meshing is obtained, and the spurious diffractions induced by staircase descriptions of boundaries are avoided. Contrary to what occurs with the vacuum method, the quality of the numerical solution obtained with this method is almost independent of the angle between the free boundary and the Cartesian meshing.