Finite Difference Scheme (finite + difference_scheme)

Distribution by Scientific Domains

Kinds of Finite Difference Scheme

  • non-standard finite difference scheme


  • Selected Abstracts


    Application of the 3D Finite Difference Scheme to the TEXTORDED Geometry

    CONTRIBUTIONS TO PLASMA PHYSICS, Issue 7-9 2006
    R. Zagórski
    First page of article [source]


    On the applicability of the HSAB principle through the use of improved computational schemes for chemical hardness evaluation

    JOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 7 2004
    Mihai V. Putz
    Abstract Finite difference schemes, named Compact Finite Difference Schemes with Spectral-like Resolution, have been used for a less crude approximation of the analytical hardness definition as the second-order derivative of the energy with respect to the electron number. The improved computational schemes, at different levels of theory, have been used to calculate global hardness values of some probe bases, traditionally classified as hard and soft on the basis of their chemical behavior, and to investigate the quantitative applicability of the HSAB principle. Exchange acid-base reactions have been used to test the HSAB principle assuming the reaction energies as a measure of the stabilization of product adducts. © 2004 Wiley Periodicals, Inc. J Comput Chem 25: 994,1003, 2004 [source]


    A visco-plastic constitutive model for granular soils modified according to non-local and gradient approaches

    INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 2 2002
    C. di Prisco
    Abstract An already available non-associated elastic,viscoplastic constitutive model with anisotropic strain hardening is modified in order to describe both the constitutive parameter dependency on relative density and the spatio-temporal evolution of strain localization. To achieve this latter goal, two distinct but similar approaches are introduced: one inspired by the gradient theory and one by the non-local theory. A one-dimensional case concerning a simple shear test for a non-homogeneous infinitely long dense sand specimen is numerically discussed and a finite difference scheme is employed for this purpose. The results obtained by following the two different approaches are critically analysed and compared. Copyright © 2001 John Wiley & Sons, Ltd. [source]


    A refined semi-analytic design sensitivity based on mode decomposition and Neumann series

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2005
    Maenghyo Cho
    Abstract Among various sensitivity evaluation techniques, semi-analytical method (SAM) is quite popular since this method is more advantageous than analytical method (AM) and global finite difference method (GFD). However, SAM reveals severe inaccuracy problem when relatively large rigid body motions are identified for individual elements. Such errors result from the pseudo load vector calculated by differentiation using the finite difference scheme. In the present study, an iterative refined semi-analytical method (IRSAM) combined with mode decomposition technique is proposed to compute reliable semi-analytical design sensitivities. The improvement of design sensitivities corresponding to the rigid body mode is evaluated by exact differentiation of the rigid body modes and the error of SAM caused by numerical difference scheme is alleviated by using a Von Neumann series approximation considering the higher order terms for the sensitivity derivatives. In eigenvalue problems, the tendency of eigenvalue sensitivity is similar to that of displacement sensitivity in static problems. Eigenvector is decomposed into rigid body mode and pure deformation mode. The present iterative SAM guarantees that the eigenvalue and eigenvector sensitivities converge to the reliable values for the wide range of perturbed size of the design variables. Accuracy and reliability of the shape design sensitivities in static problems and eigenvalue problems by the proposed method are assessed through the various numerical examples. Copyright © 2004 John Wiley & Sons, Ltd. [source]


    An unconditionally stable three level finite difference scheme for solving parabolic two-step micro heat transport equations in a three-dimensional double-layered thin film

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004
    Weizhong Dai
    Abstract Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equations are parabolic two-step equations, which are different from the traditional heat diffusion equation. In this study, we develop a three-level finite difference scheme for solving the micro heat transport equations in a three-dimensional double-layered thin film. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results for thermal analysis of a gold layer on a chromium padding layer are obtained. Copyright © 2003 John Wiley & Sons, Ltd. [source]


    Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2003
    N. SukumarArticle first published online: 11 MAR 200
    Abstract Voronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids. Natural neighbours are based on the Voronoi diagram, which partitions space into closest-point regions. The Sibson and the Laplace (non-Sibsonian) interpolants which are based on natural neighbours have shown promise within a Galerkin framework for the solution of partial differential equations. In this paper, we focus on the Laplace interpolant with a two-fold objective: first, to unify the previous developments related to the Laplace interpolant and to indicate its ties to some well-known numerical methods; and secondly to propose a Voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids. A conservation law in integral form is discretized on Voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids. The proposed scheme can also be viewed as a point collocation technique. A detailed study on consistency is conducted, and the satisfaction of the discrete maximum principle (stability) is established. Owing to symmetry of the Laplace weight, a symmetric positive-definite stiffness matrix is realized which permits the use of efficient linear solvers. On a regular (rectangular or hexagonal) grid, the difference scheme reduces to the classical finite difference method. Numerical examples for the Poisson equation with Dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme. Copyright © 2003 John Wiley & Sons, Ltd. [source]


    Composite high resolution localized relaxation scheme based on upwinding for hyperbolic conservation laws

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2009
    Ritesh Kumar Dubey
    Abstract In this work we present an upwind-based high resolution scheme using flux limiters. Based on the direction of flow we choose the smoothness parameter in such a way that it leads to a truly upwind scheme without losing total variation diminishing (TVD) property for hyperbolic linear systems where characteristic values can be of either sign. Here we present and justify the choice of smoothness parameters. The numerical flux function of a high resolution scheme is constructed using wave speed splitting so that it results into a scheme that truly respects the physical hyperbolicity property. Bounds are given for limiter functions to satisfy TVD property. The proposed scheme is extended for non-linear problems by using the framework of relaxation system that converts a non-linear conservation law into a system of linear convection equations with a non-linear source term. The characteristic speed of relaxation system is chosen locally on three point stencil of grid. This obtained relaxation system is solved using composite scheme technique, i.e. using a combination of proposed scheme with the conservative non-standard finite difference scheme. Presented numerical results show higher resolution near discontinuity without introducing spurious oscillations. Copyright © 2008 John Wiley & Sons, Ltd. [source]


    Depth-integrated, non-hydrostatic model for wave breaking and run-up

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2009
    Yoshiki Yamazaki
    Abstract This paper describes the formulation, verification, and validation of a depth-integrated, non-hydrostatic model with a semi-implicit, finite difference scheme. The formulation builds on the nonlinear shallow-water equations and utilizes a non-hydrostatic pressure term to describe weakly dispersive waves. A momentum-conserved advection scheme enables modeling of breaking waves without the aid of analytical solutions for bore approximation or empirical equations for energy dissipation. An upwind scheme extrapolates the free-surface elevation instead of the flow depth to provide the flux in the momentum and continuity equations. This greatly improves the model stability, which is essential for computation of energetic breaking waves and run-up. The computed results show very good agreement with laboratory data for wave propagation, transformation, breaking, and run-up. Since the numerical scheme to the momentum and continuity equations remains explicit, the implicit non-hydrostatic solution is directly applicable to existing nonlinear shallow-water models. Copyright © 2008 John Wiley & Sons, Ltd. [source]


    Splitting methods for high order solution of the incompressible Navier,Stokes equations in 3D

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005
    Arnim Brüger
    Abstract The incompressible Navier,Stokes equations are discretized in space by a hybrid method and integrated in time by the method of lines. The solution is determined on a staggered curvilinear grid in two space dimensions and by a Fourier expansion in the third dimension. The space derivatives are approximated by a compact finite difference scheme of fourth-order on the grid. The solution is advanced in time by a semi-implicit method. In each time step, systems of linear equations have to be solved for the velocity and the pressure. The iterations are split into one outer iteration and three inner iterations. The accuracy and efficiency of the method are demonstrated in a numerical experiment with rotated Poiseuille flow perturbed by Orr,Sommerfeld modes in a channel. Copyright © 2005 John Wiley & Sons, Ltd. [source]


    High-order boundary conditions for linearized shallow water equations with stratification, dispersion and advection,

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2004
    Vince J. van Joolen
    Abstract The two-dimensional linearized shallow water equations are considered in unbounded domains with density stratification. Wave dispersion and advection effects are also taken into account. The infinite domain is truncated via a rectangular artificial boundary ,, and a high-order open boundary condition (OBC) is imposed on ,. Then the problem is solved numerically in the finite domain bounded by ,. A recently developed boundary scheme is employed, which is based on a reformulation of the sequence of OBCs originally proposed by Higdon. The OBCs can easily be used up to any desired order. They are incorporated here in a finite difference scheme. Numerical examples are used to demonstrate the performance and advantages of the computational method, with an emphasis is on the effect of stratification. Published in 2004 by John Wiley & Sons, Ltd. [source]


    An efficient finite difference scheme for free-surface flows in narrow rivers and estuaries

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2003
    XinJian ChenArticle first published online: 13 MAY 200
    Abstract This paper presents a free-surface correction (FSC) method for solving laterally averaged, 2-D momentum and continuity equations. The FSC method is a predictor,corrector scheme, in which an intermediate free surface elevation is first calculated from the vertically integrated continuity equation after an intermediate, longitudinal velocity distribution is determined from the momentum equation. In the finite difference equation for the intermediate velocity, the vertical eddy viscosity term and the bottom- and sidewall friction terms are discretized implicitly, while the pressure gradient term, convection terms, and the horizontal eddy viscosity term are discretized explicitly. The intermediate free surface elevation is then adjusted by solving a FSC equation before the intermediate velocity field is corrected. The finite difference scheme is simple and can be easily implemented in existing laterally averaged 2-D models. It is unconditionally stable with respect to gravitational waves, shear stresses on the bottom and side walls, and the vertical eddy viscosity term. It has been tested and validated with analytical solutions and field data measured in a narrow, riverine estuary in southwest Florida. Model simulations show that this numerical scheme is very efficient and normally can be run with a Courant number larger than 10. It can be used for rivers where the upstream bed elevation is higher than the downstream water surface elevation without any problem. Copyright © 2003 John Wiley & Sons, Ltd. [source]


    Highly accurate solutions of the bifurcation structure of mixed-convection heat transfer using spectral method

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2002
    M. Selmi
    Abstract This paper is concerned with producing highly accurate solution and bifurcation structure using the pseudo-spectral method for the two-dimensional pressure-driven flow through a horizontal duct of a square cross-section that is heated by a uniform flux in the axial direction with a uniform temperature on the periphery. Two approaches are presented. In one approach, the streamwise vorticity, streamwise momentum and energy equations are solved for the stream function, axial velocity, and temperature. In the second approach, the streamwise vorticity and a combination of the energy and momentum equations are solved for stream function and temperature only. While the second approach solves less number of equations than the first approach, a grid sensitivity analysis has shown no distinct advantage of one method over the other. The overall solution structure composed of two symmetric and four asymmetric branches in the range of Grashof number (Gr) of 0,2 × 106 for a Prandtl number (Pr) of 0.73 has been computed using the first approach. The computed structure is comparable to that found by Nandakumar and Weinitschke (1991) using a finite difference scheme for Grashof numbers in the range of 0,1×106. The stability properties of some solution branches; however, are different. In particular, the two-cell structure of the isolated symmetric branch that has been found to be unstable by the study of Nandakumar and Weinitschke is found to be stable by the current study. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    A new parallelization strategy for solving time-dependent 3D Maxwell equations using a high-order accurate compact implicit scheme,

    INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 5 2006
    Eugene Kashdan
    Abstract With progress in computer technology there has been renewed interest in a time-dependent approach to solving Maxwell equations. The commonly used Yee algorithm (an explicit central difference scheme for approximation of spatial derivatives coupled with the Leapfrog scheme for approximation of temporal derivatives) yields only a second-order of accuracy. On the other hand, an increasing number of industrial applications, especially in optic and microwave technology, demands high-order accurate numerical modelling. The standard way to increase accuracy of the finite difference scheme without increasing the differential stencil is to replace a 2nd-order accurate explicit scheme for approximation of spatial derivatives with the 4th-order accurate compact implicit scheme. In general, such a replacement requires additional memory resources and slows the computations. However, the curl-based form of Maxwell equations allows us to construct an effective parallel algorithm with the alternating domain decomposition (ADD) minimizing the communication time. We present a new parallel approach to the solution of three-dimensional time-dependent Maxwell equations and provide a theoretical and experimental analysis of its performance. Copyright © 2006 John Wiley & Sons, Ltd. [source]


    On an initial-boundary value problem for a wide-angle parabolic equation in a waveguide with a variable bottom

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2009
    V. A. Dougalis
    Abstract We consider the third-order Claerbout-type wide-angle parabolic equation (PE) of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range-dependent topography. There is strong indication that the initial-boundary value problem for this equation with just a homogeneous Dirichlet boundary condition posed on B may not be well-posed, for example when B is downsloping. We impose, in addition to the above, another homogeneous, second-order boundary condition, derived by assuming that the standard (narrow-angle) PE holds on B, and establish a priori H2 estimates for the solution of the resulting initial-boundary value problem for any bottom topography. After a change of the depth variable that makes B horizontal, we discretize the transformed problem by a second-order accurate finite difference scheme and show, in the case of upsloping and downsloping wedge-type domains, that the new model gives stable and accurate results. We also present an alternative set of boundary conditions that make the problem exactly energy conserving; one of these conditions may be viewed as a generalization of the Abrahamsson,Kreiss boundary condition in the wide-angle case. Copyright © 2008 John Wiley & Sons, Ltd. [source]


    UNSTEADY STATE DISPERSION OF AIR POLLUTANTS UNDER THE EFFECTS OF DELAYED AND NONDELAYED REMOVAL MECHANISMS

    NATURAL RESOURCE MODELING, Issue 4 2009
    MANJU AGARWAL
    Abstract In this paper, we present a two-dimensional time-dependent mathematical model for studying the unsteady state dispersion of air pollutants emitted from an elevated line source in the atmosphere under the simultaneous effects of delayed (slow) and nondelayed (instantaneous) removal mechanisms. The wind speed and coefficient of diffusion are taken as functions of the vertical height above the ground. The deposition of pollutants on the absorptive ground and leakage into the atmosphere at the inversion layer are also included in the model by applying appropriate boundary conditions. The model is solved numerically by the fractional step method. The Lagrangian approach is used to solve the advection part, whereas the Eulerian finite difference scheme is applied to solve the part with the diffusion and removal processes. The solutions are analyzed to observe the effects of coexisting delayed and nondelayed removal mechanisms on overall dispersion. Comparison of delayed and nondelayed removal processes of equal capacity shows that the latter (nondelayed) process is more effective than the former (delayed removal) in the removal of pollutants from the atmosphere. [source]


    A space-time mixed-hybrid finite element method for the damped wave equation

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2008
    A. Serghini Mounim
    Abstract A space-time finite element method is introduced to solve the linear damped wave equation. The scheme is constructed in the framework of the mixed-hybrid finite element methods, and where an original conforming approximation of H(div;,) is used, the latter permits us to obtain an upwind scheme in time. We establish the link between the nonstandard finite difference scheme recently introduced by Mickens and Jordan and the scheme proposed. In this regard, two approaches are considered and in particular we employ a formulation allowing the solution to be marched in time, i.e., one only needs to consider one time increment at a time. Numerical results are presented and compared with the analytical solution illustrating good performance of the present method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 [source]


    Alternating direction finite volume element methods for 2D parabolic partial differential equations

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2008
    Tongke Wang
    Abstract On the basis of rectangular partition and bilinear interpolation, this article presents alternating direction finite volume element methods for two dimensional parabolic partial differential equations and gives three computational schemes, one is analogous to Douglas finite difference scheme with second order splitting error, the second has third order splitting error, and the third is an extended locally one dimensional scheme. Optimal L2 norm or H1 semi-norm error estimates are obtained for these schemes. Finally, two numerical examples illustrate the effectiveness of the schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source]


    A convergent three-level finite difference scheme for solving a dual-phase-lagging heat transport equation in spherical coordinates

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2004
    Weizhong Dai
    Abstract Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in spherical coordinates and develop a three-level finite difference scheme for solving the heat transport equation in a microsphere. It is shown that the scheme is convergent, which implies that the scheme is unconditionally stable. Results show that the numerical solution converges to the exact solution. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 60,71, 2004. [source]


    An unconditionally stable and O(,2 + h4) order L, convergent difference scheme for linear parabolic equations with variable coefficients

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2001
    Zhi-Zhong Sun
    Abstract M. K. Jain, R. K. Jain, and R. K. Mohanty presented a finite difference scheme of O(,2 + ,h2 + h4) for solving the one-dimensional quasilinear parabolic partial differential equation, uxx = f(x, t, u, ut, ux), with Dirichlet boundary conditions. The method, when applied to a linear constant coefficient case, was shown to be unconditionally stable by the Von Neumann method. In this article, we prove that the method, when applied to a linear variable coefficient case, is unconditionally stable and convergent with the convergence order O(,2 + h4) in the L, -norm. In addition, we obtain an asymptotic expansion of the difference solution, with which we obtain an O(,4 + ,2h4 + h6) order accuracy approximation after extrapolation. And last, we point out that the analysis method in this article is efficacious for complex equations. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:619,631, 2001 [source]


    Low-curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes

    COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2004
    Andrea L. Bertozzi
    We consider a class of fourth-order nonlinear diffusion equations motivated by Tumblin and Turk's "low-curvature image simplifiers" for image denoising and segmentation. The PDE for the image intensity u is of the form where g(s) = k2/(k2 + s2) is a "curvature" threshold and , denotes a fidelity-matching parameter. We derive a priori bounds for ,u that allow us to prove global regularity of smooth solutions in one space dimension, and a geometric constraint for finite-time singularities from smooth initial data in two space dimensions. This is in sharp contrast to the second-order Perona-Malik equation (an ill-posed problem), on which the original LCIS method is modeled. The estimates also allow us to design a finite difference scheme that satisfies discrete versions of the estimates, in particular, a priori bounds on the smoothness estimator in both one and two space dimensions. We present computational results that show the effectiveness of such algorithms. Our results are connected to recent results for fourth-order lubrication-type equations and the design of positivity-preserving schemes for such equations. This connection also has relevance for other related fourth-order imaging equations. © 2004 Wiley Periodicals, Inc. [source]


    Computational methods for optical molecular imaging

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2009
    Duan Chen
    Abstract A new computational technique, the matched interface and boundary (MIB) method, is presented to model the photon propagation in biological tissue for the optical molecular imaging. Optical properties have significant differences in different organs of small animals, resulting in discontinuous coefficients in the diffusion equation model. Complex organ shape of small animal induces singularities of the geometric model as well. The MIB method is designed as a dimension splitting approach to decompose a multidimensional interface problem into one-dimensional ones. The methodology simplifies the topological relation near an interface and is able to handle discontinuous coefficients and complex interfaces with geometric singularities. In the present MIB method, both the interface jump condition and the photon flux jump conditions are rigorously enforced at the interface location by using only the lowest-order jump conditions. This solution near the interface is smoothly extended across the interface so that central finite difference schemes can be employed without the loss of accuracy. A wide range of numerical experiments are carried out to validate the proposed MIB method. The second-order convergence is maintained in all benchmark problems. The fourth-order convergence is also demonstrated for some three-dimensional problems. The robustness of the proposed method over the variable strength of the linear term of the diffusion equation is also examined. The performance of the present approach is compared with that of the standard finite element method. The numerical study indicates that the proposed method is a potentially efficient and robust approach for the optical molecular imaging. Copyright © 2008 John Wiley & Sons, Ltd. [source]


    Matched interface and boundary (MIB) method for the vibration analysis of plates

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 9 2009
    S. N. Yu
    Abstract This paper proposes a novel approach, the matched interface and boundary (MIB) method, for the vibration analysis of rectangular plates with simply supported, clamped and free edges, and their arbitrary combinations. In previous work, the MIB method was developed for three-dimensional elliptic equations with arbitrarily complex material interfaces and geometric shapes. The present work generalizes the MIB method for eigenvalue problems in structural analysis with complex boundary conditions. The MIB method utilizes both uniform and non-uniform Cartesian grids. Fictitious values are utilized to facilitate the central finite difference schemes throughout the entire computational domain. Boundary conditions are enforced with fictitious values,a common practice used in the previous discrete singular convolution algorithm. An essential idea of the MIB method is to repeatedly use the boundary conditions to achieve arbitrarily high-order accuracy. A new feature in the proposed approach is the implementation of the cross derivatives in the free boundary conditions. The proposed method has a banded matrix. Nine different plates, particularly those with free edges and free corners, are employed to validate the proposed method. The performance of the proposed method is compared with that of other established methods. Convergence and comparison studies indicate that the proposed MIB method works very well for the vibration analysis of plates. In particular, modal bending moments and shear forces predicted by the proposed method vanish at boundaries for free edges. Copyright © 2008 John Wiley & Sons, Ltd. [source]


    Numerical analysis of Rayleigh,Plesset equation for cavitating water jets

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2007
    H. Alehossein
    Abstract High-pressure water jets are used to cut and drill into rocks by generating cavitating water bubbles in the jet which collapse on the surface of the rock target material. The dynamics of submerged bubbles depends strongly on the surrounding pressure, temperature and liquid surface tension. The Rayleigh,Plesset (RF) equation governs the dynamic growth and collapse of a bubble under various pressure and temperature conditions. A numerical finite difference model is established for simulating the process of growth, collapse and rebound of a cavitation bubble travelling along the flow through a nozzle producing a cavitating water jet. A variable time-step technique is applied to solve the highly non-linear second-order differential equation. This technique, which emerged after testing four finite difference schemes (Euler, central, modified Euler and Runge,Kutta,Fehlberg (RKF)), successfully solves the Rayleigh,Plesset (RP) equation for wide ranges of pressure variation and bubble initial sizes and saves considerable computing time. Inputs for this model are the pressure and velocity data obtained from a CFD (computational fluid dynamics) analysis of the jet. Copyright © 2007 John Wiley & Sons, Ltd. [source]


    Development of a class of multiple time-stepping schemes for convection,diffusion equations in two dimensions

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006
    R. K. Lin
    Abstract In this paper we present a class of semi-discretization finite difference schemes for solving the transient convection,diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection,diffusion (CD) equation to the inhomogeneous steady convection,diffusion-reaction (CDR) equation after using different time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed time-stepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection,diffusion-reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source]


    On the influence of numerical schemes and subgrid,stress models on large eddy simulation of turbulent flow past a square cylinder

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2002
    A. Nakayama
    Abstract Influence of finite difference schemes and subgrid-stress models on the large eddy simulation calculation of turbulent flow around a bluff body of square cylinder at a laboratory Reynolds number, has been examined. It is found that the type and the order of accuracy of finite-difference schemes and the subgrid-stress model for satisfactory results are dependent on each other, and the grid resolution and the Reynolds number. Using computational grids manageable by workstation-level computers, with which the near-wall region of the separating boundary layer cannot be resolved, central-difference schemes of realistic orders of accuracy, either fully conservative or non-conservative, suffer stability problems. The upwind-biased schemes of third order and the Smagorinsky eddy-viscosity subgrid model can give reasonable results resolving much of the energy-containing turbulent eddies in the boundary layers and in the wake and representing the subgrid stresses in most parts of the flow. Noticeable improvements can be obtained by either using higher order difference schemes, increasing the grid resolution and/or by implementing a dynamic subgrid stress model, but each at a cost of increased computational time. For further improvements, the very small-scale eddies near the upstream corners and in the laminar sublayers need to be resolved but would require a substantially larger number of grid points that are out of the range of easily accessible computers. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    Transition probability coefficients and the stability of finite difference schemes for the diffusion and Telegraphers' equations

    INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 3 2002
    Michal J. Malachowski
    A probabilistic approach has been used to analyse the stability of the various finite difference formulations for propagation of signals on a lossy transmission line. If the sign of certain transition probabilities is negative, then the algorithm is found to be unstable. We extend the concept to consider the effects of space and time discretizations on the signs of the coefficients in a probabilistic finite difference implementation of the Telegraphers' equation and draw parallels with the transmission line matrix (TLM) technique. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    Non-standard finite difference schemes for multi-dimensional second-order systems in non-smooth mechanics

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2007
    Yves Dumont
    Abstract This work is an extension of the paper (Proc. R. Soc. London 2005; 461A:1927,1950) to impact oscillators with more than one degree of freedom. Given the complex and even chaotic behaviour of these non-smooth mechanical systems, it is essential to incorporate their qualitative physical properties, such as the impact law and the frequencies of the systems, into the envisaged numerical methods if the latter is to be reliable. Based on this strategy, we design several non-standard finite difference schemes. Apart from their excellent error bounds and unconditional stability, the schemes are analysed for their efficiency to preserve some important physical properties of the systems including, among others, the conservation of energy between consecutive impact times, the periodicity of the motion and the boundedness of the solutions. Numerical simulations that support the theory are provided. Copyright © 2006 John Wiley & Sons, Ltd. [source]


    Numerical simulations of the improved Boussinesq equation

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2010
    Dursun Irk
    Abstract In this study, numerical simulations of the improved Boussinesq equation are obtained using two finite difference schemes and two finite element methods, based on the second-and third-order time discretization. The methods are tested on the problems of propagation of a soliton and interaction of two solitons. After the L, error norm is used to measure differences between the exact and numerical solutions, the results obtained by the proposed methods are compared with recently published results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source]


    Numerical studies of a nonlinear heat equation with square root reaction term

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2009
    Ron Buckmire
    Abstract Interest in calculating numerical solutions of a highly nonlinear parabolic partial differential equation with fractional power diffusion and dissipative terms motivated our investigation of a heat equation having a square root nonlinear reaction term. The original equation occurs in the study of plasma behavior in fusion physics. We begin by examining the numerical behavior of the ordinary differential equation obtained by dropping the diffusion term. The results from this simpler case are then used to construct nonstandard finite difference schemes for the partial differential equation. A variety of numerical results are obtained and analyzed, along with a comparison to the numerics of both standard and several nonstandard schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]


    Energy properties preserving schemes for Burgers' equation,

    NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2008
    R. Anguelov
    Abstract The Burgers' equation, a simplification of the Navier,Stokes equations, is one of the fundamental model equations in gas dynamics, hydrodynamics, and acoustics that illustrates the coupling between convection/advection and diffusion. The kinetic energy enjoys boundedness and monotone decreasing properties that are useful in the study of the asymptotic behavior of the solution. We construct a family of non-standard finite difference schemes, which replicate the energy equality and the properties of the kinetic energy. Our approach is based on Mickens' rule [Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.] of nonlocal approximation of nonlinear terms. More precisely, we propose a systematic nonlocal way of generating approximations that ensure that the trilinear form is identically zero for repeated arguments. We provide numerical experiments that support the theory and demonstrate the power of the non-standard schemes over the classical ones. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source]