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Finite Difference Methods (finite + difference_methods)
Selected AbstractsSome finite difference methods for a kind of GKdV equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2007X. Lai Abstract In this paper, some finite difference schemes I, II, III and IV, are investigated and compared in solving a kind of mixed problem of generalized Korteweg-de Vries (GKdV) equations especially the relative errors. Both the numerical dispersion and the numerical dissipation are analysed for the constructed difference scheme I. The stability is also obtained for scheme I and the constructed predictor,corrector scheme IV by using a linearized stability method. Other two schemes, II and III, are also included in the comparison among these four schemes for the numerical analysis of different GKdV equations. The results enable one to consider the relative error when dealing with these kinds of GKdV equations. Copyright © 2006 John Wiley & Sons, Ltd. [source] Error estimates in 2-node shear-flexible beam elementsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2003Gajbir Singh Abstract The objective of the paper is to report the investigation of error estimates/or convergence characteristics of shear-flexible beam elements. The order and magnitude of principal discretization error in the usage of various types beam elements such as: (a) 2-node standard isoparametric element, (b) 2-node field-consistent/reduced integration element and (c) 2-node coupled-displacement field element, is assessed herein. The method employs classical order of error analyses that is commonly used to evaluate the discretization error of finite difference methods. The finite element equilibrium equations at any node are expressed in terms of differential equations through the use of Taylor series. These differential equations are compared with the governing equations and error terms are identified. It is shown that the discretization error in coupled-field elements is the least compared to the field-consistent and standard finite elements (based on exact integration). Copyright © 2003 John Wiley & Sons, Ltd. [source] A partition-of-unity-based finite element method for level setsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2008Stéphane Valance Abstract Level set methods have recently gained much popularity to capture discontinuities, including their possible propagation. Typically, the partial differential equations that arise in level set methods, in particular the Hamilton,Jacobi equation, are solved by finite difference methods. However, finite difference methods are less suited for irregular domains. Moreover, it seems slightly awkward to use finite differences for the capturing of a discontinuity, while in a subsequent stress analysis finite elements are normally used. For this reason, we here present a finite element approach to solving the governing equations of level set methods. After a review of the governing equations, the initialization of the level sets, the discretization on a finite domain, and the stabilization of the resulting finite element method will be discussed. Special attention will be given to the proper treatment of the internal boundary condition, which is achieved by exploiting the partition-of-unity property of finite element shape functions. Finally, a quantitative analysis including accuracy analysis is given for a one-dimensional example and a qualitative example is given for a two-dimensional case with a curved discontinuity. Copyright © 2008 John Wiley & Sons, Ltd. [source] Comparing vortex methods and finite difference methods in a homogeneous turbulent shear flowINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2010R. Yokota Abstract The vortex method is applied to the calculation of a homogeneous shear turbulence, and compared with a finite difference code using identical calculation conditions. The core spreading method with spatial adaptation is selected as the viscous diffusion scheme of the vortex method. The shear rate is chosen so that it matches the maximum value observed in a fully developed channel flow. The isosurface, anisotropy tensors, and joint probability density functions reflect the ability of the present vortex method to quantitatively reproduce the anisotropic nature of strongly sheared turbulence, both instantaneously and statistically. Copyright © 2009 John Wiley & Sons, Ltd. [source] Sensitivity computations of eddy viscosity models with an application in drag computationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2006Faranak PahlevaniArticle first published online: 10 FEB 200 Abstract This paper presents a numerical study of the sensitivity of an eddy viscosity model with respect to the variation of the eddy viscosity parameter for the two-dimensional driven cavity problem and flow around a cylinder. The main objective is to provide a comparison between computing the sensitivity using sensitivity equation and computing the sensitivity using finite difference methods and also numerically illustrate the application of the sensitivity computations in improving drag flow functional. Copyright © 2006 John Wiley & Sons, Ltd. [source] Non-linear additive Schwarz preconditioners and application in computational fluid dynamicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2002Xiao-Chuan Cai Abstract The focus of this paper is on the numerical solution of large sparse non-linear systems of algebraic equations on parallel computers. Such non-linear systems often arise from the discretization of non-linear partial differential equations, such as the Navier,Stokes equations for fluid flows, using finite element or finite difference methods. A traditional inexact Newton method, applied directly to the discretized system, does not work well when the non-linearities in the algebraic system become unbalanced. In this paper, we study some preconditioned inexact Newton algorithms, including the single-level and multilevel non-linear additive Schwarz preconditioners. Some results for solving the high Reynolds number incompressible Navier,Stokes equations are reported. Copyright © 2002 John Wiley & Sons, Ltd. [source] Numerical simulation of the vertical structure of discontinuous flowsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2001Guus S. Stelling Abstract A numerical method to solve the Reynolds-averaged Navier,Stokes equations with the presence of discontinuities is outlined and discussed. The pressure is decomposed into the sum of a hydrostatic component and a hydrodynamic component. The numerical technique is based upon the classical staggered grids and semi-implicit finite difference methods applied for quasi- and non-hydrostatic flows. The advection terms in the momentum equations are approximated in order to conserve mass and momentum following the principles recently developed for the numerical simulation of shallow water flows with large gradients. Conservation of these properties is the most important aspect to represent near local discontinuities in the solution, following from sharp bottom gradients or hydraulic jumps. The model is applied to reproduce the flow over a step where a hydraulic jump forms downstream. The hydrostatic pressure assumption fails to represent this type of flow mainly because of the pressure deviation from the hydrostatic values downstream the step. Fairly accurate results are obtained from the numerical model compared with experimental data. Deviation from the data is found to be inherent to the standard k,, model implemented. Copyright © 2001 John Wiley & Sons, Ltd. [source] Sorption dynamics in fixed-beds of inert core spherical adsorbents including axial dispersion and Langmuir isothermAICHE JOURNAL, Issue 7 2009M. Khosravi Koocheksarayi Abstract The effects of axial dispersion and Langmuir isotherm on transient behavior of sorption and intraparticle diffusion in fixed-beds packed with monodisperse shell-type/inert core spherical sorbents are studied. The system of partial differential equations of the mathematical model is solved numerically using finite difference methods. Results are presented in the form of breakthrough curves for adsorption and desorption processes. Results reveal that the shape of the breakthrough curves is influenced by both hydrodynamic and kinetic factors. Hydrodynamic factor is governed by axial dispersion and is controlled by changes of Peclet number. Simulation results reveal that when linear adsorption isotherm is used, the effect of axial dispersion on breakthrough curves of the system is important for Peclet numbers smaller than 50, whereas, for Langmuir isotherm axial dispersion is considerable for Peclet numbers less than 80. In addition, effects of type of adsorption isotherms and size of adsorbents on breakthrough curves are investigated, and results are compared with existing reports in the pertinent literature. © 2009 American Institute of Chemical Engineers AIChE J, 2009 [source] Solving singularly perturbed advection,reaction equations via non-standard finite difference methodsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2007Jean M.-S. Abstract We design and implement two non-standard finite difference methods (NSFDMs) to solve singularly perturbed advection,reaction equations (SPARE). Our methods constitute a big plus to the class of those ,rare' fitted operator methods, which can be extended to singularly perturbed partial differential equations. Unlike the standard finite difference methods (SFDMs), the NSFDMs designed in this paper allow the time and the space step sizes to vary independently of one another and of the parameter , in the SPARE under consideration. The NSFDMs replicate the linear stability properties of the fixed points of the continuous problem. Furthermore, these methods preserve the positivity and boundedness properties of the exact solution. Numerical simulations that confirm the theoretical results are presented. Copyright © 2007 John Wiley & Sons, Ltd. [source] Performance and numerical behavior of the second-order scheme of precise time-step integration for transient dynamic analysisNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2007Hang Ma Abstract Spurious high-frequency responses resulting from spatial discretization in time-step algorithms for structural dynamic analysis have long been an issue of concern in the framework of traditional finite difference methods. Such algorithms should be not only numerically dissipative in a controllable manner, but also unconditionally stable so that the time-step size can be governed solely by the accuracy requirement. In this article, the issue is considered in the framework of the second-order scheme of the precise integration method (PIM). Taking the Newmark-, method as a reference, the performance and numerical behavior of the second-order PIM for elasto-dynamic impact-response problems are studied in detail. In this analysis, the differential quadrature method is used for spatial discretization. The effects of spatial discretization, numerical damping, and time step on solution accuracy are explored by analyzing longitudinal vibrations of a shock-excited rod with rectangular, half-triangular, and Heaviside step impact. Both the analysis and numerical tests show that under the framework of the PIM, the spatial discretization used here can provide a reasonable number of model types for any given error tolerance. In the analysis of dynamic response, an appropriate spatial discretization scheme for a given structure is usually required in order to obtain an accurate and meaningful numerical solution, especially for describing the fine details of traction responses with sharp changes. Under the framework of the PIM, the numerical damping that is often required in traditional integration schemes is found to be unnecessary, and there is no restriction on the size of time steps, because the PIM can usually produce results with machine-like precision and is an unconditionally stable explicit method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation,NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2005Mehdi Dehghan Abstract Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second-order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one-dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source] Coupling finite difference methods and integral formulas for elliptic problems arising in fluid mechanicsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2004C. Albuquerque Abstract This article is devoted to the numerical analysis of two classes of iterative methods that combine integral formulas with finite-difference Poisson solvers for the solution of elliptic problems. The first method is in the spirit of the Schwarz domain decomposition method for exterior domains. The second one is motivated by potential calculations in free boundary problems and can be viewed as a numerical analytic continuation algorithm. Numerical tests are presented that confirm the convergence properties predicted by numerical analysis. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 199,229, 2004 [source] The accuracy and efficiency of alternative option pricing approaches relative to a log-transformed trinomial modelTHE JOURNAL OF FUTURES MARKETS, Issue 6 2002Hsuan-Chi Chen This article presents a log-transformed trinomial approach to option pricing and finds that various numerical procedures in the option pricing literature are embedded in this approach with choices of different parameters. The unified view also facilitates comparisons of computational efficiency among numerous lattice approaches and explicit finite difference methods. We use the root-mean-squared relative error and the minimum convergence step to evaluate the accuracy and efficiency for alternative option pricing approaches. The numerical results show that the equal-probability trinomial specification of He (12) and Tian (25) and the sharpened trinomial specification of Omberg (21) outperform other lattice approaches and explicit finite difference methods. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:557,577, 2002 [source] | |||||||||||||