scispace - formally typeset
Search or ask a question

Showing papers on "Finite difference method published in 2003"


Journal ArticleDOI
TL;DR: In this article, a local radial basis function-based differential quadrature (LRQ) method is proposed, which discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots, which may be distributed randomly.

475 citations


Journal ArticleDOI
TL;DR: This book surveys higher-order finite difference methods and develops various mass-lumped finite element methods for the transient wave equations, and presents the most efficient methods, respecting both accuracy and stability for each sort of problem.
Abstract: Solving efficiently the wave equations involved in modeling acoustic, elastic or electromagnetic wave propagation remains a challenge both for research and industry. To attack the problems coming from the propagative character of the solution, the author constructs higher-order numerical methods to reduce the size of the meshes, and consequently the time and space stepping, dramatically improving storage and computing times. This book surveys higher-order finite difference methods and develops various mass-lumped finite (also called spectral) element methods for the transient wave equations, and presents the most efficient methods, respecting both accuracy and stability for each sort of problem. A central role is played by the notion of the dispersion relation for analyzing the methods. The last chapter is devoted to unbounded domains which are modeled using perfectly matched layer (PML) techniques. Numerical examples are given.

464 citations


Journal ArticleDOI
TL;DR: New finite element methods based on Cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients, and these new methods can be used as finite difference methods.
Abstract: New finite element methods based on Cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients. The triangulations in these methods do not need to fit the interfaces. The basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately. Both non-conforming and conforming finite element spaces are considered. Corresponding interpolation functions are proved to be second order accurate in the maximum norm. The conforming finite element method has been shown to be convergent. With Cartesian triangulations, these new methods can be used as finite difference methods. Numerical examples are provided to support the methods and the theoretical analysis.

425 citations


Journal ArticleDOI
TL;DR: In this article, a virtual boundary method is applied to the numerical simulation of a uniform flow over a cylinder, where the immersed boundary is represented with a finite number of Lagrangian points, distributed over the solid-fluid interface.

366 citations


Journal ArticleDOI
TL;DR: The results of computations with eight explicit finite difference schemes on a suite of one-dimensional and two-dimensional test problems for the Euler equations are presented in various formats.
Abstract: The results of computations with eight explicit finite difference schemes on a suite of one-dimensional and two-dimensional test problems for the Euler equations are presented in various formats. Both dimensionally split and two-dimensional schemes are represented, as are central and upwind-biased methods, and all are at least second-order accurate.

365 citations


Book
26 Jun 2003
TL;DR: The Finite Difference Method for the Poisson Equation and the Finite Element Method for Linear Elliptic Boundary Value Problems of Second Order have been proposed in this article.
Abstract: For Example: Modelling Processes in Porous Media with Differential Equations.- For the Beginning: The Finite Difference Method for the Poisson Equation.- The Finite Element Method for the Poisson Equation.- The Finite Element Method for Linear Elliptic Boundary Value Problems of Second Order.- Grid Generation and a posteriori Error Estimation.- Iterative Methods for Systems of Linear Equations.- The Finite Volume Method.- Discretization Methods for Parabolic Initial Boundary Value Problems.- Iterative Methods for Nonlinear Equations.- Discretization Methods for Convection-Dominated Problems.- Appendices.

278 citations


Journal ArticleDOI
TL;DR: A way of using RBF as the basis for PDE’s solvers is presented, its essence being constructing approximate formulas for derivatives discretizations based on RBF interpolants with local supports similar to stencils in finite difference methods.
Abstract: A way of using RBF as the basis for PDE’s solvers is presented, its essence being constructing approximate formulas for derivatives discretizations based on RBF interpolants with local supports similar to stencils in finite difference methods. Numerical results for different types of elasticity equations showing reasonable accuracy and good h-convergence properties of the technique are presented. In particular, examples of RBF solution in the case of non-linear Karman-Fopple equations are considered.

239 citations


Journal ArticleDOI
TL;DR: In this article, a new non-reflecting boundary scheme is proposed for time-dependent wave problems in unbounded domains, which is based on a reformulation of the sequence of NRBCs proposed by Higdon.

222 citations


Journal ArticleDOI
TL;DR: An Eulerian framework for computing the thickness of tissues between two simply connected boundaries that does not require landmark points or parameterizations of either boundary is outlined and shows strong potential for application in tissue thickness visualization and quantification.
Abstract: We outline an Eulerian framework for computing the thickness of tissues between two simply connected boundaries that does not require landmark points or parameterizations of either boundary. Thickness is defined as the length of correspondence trajectories, which run from one tissue boundary to the other, and which follow a smooth vector field constructed in the region between the boundaries. A pair of partial differential equations (PDEs) that are guided by this vector field are then solved over this region, and the sum of their solutions yields the thickness of the tissue region. Unlike other approaches, this approach does not require explicit construction of any correspondence trajectories. An efficient, stable, and computationally fast solution to these PDEs is found by careful selection of finite differences according to an up-winding condition. The behavior and performance of our method is demonstrated on two simulations and two magnetic resonance imaging data sets in two and three dimensions. These experiments reveal very good performance and show strong potential for application in tissue thickness visualization and quantification.

201 citations


Journal ArticleDOI
TL;DR: In this article, a finite-difference scheme for the electromagnetic field in 3D anisotropic media for electromagnetic logging was proposed, which has the following features: coercivity (i.e., the complete discrete analogy of all continuous equations in every grid cell, even for nondiagonal conductivity tensors), a special conductivity averaging, and a spectrally optimal grid refinement minimizing the error at the receiver locations and optimizing the approximation of the boundary conditions at infinity.
Abstract: We consider a problem of computing the electromagnetic field in 3D anisotropic media for electromagnetic logging. The proposed finite-difference scheme for Maxwell equations has the following new features based on some recent and not so recent developments in numerical analysis: coercivity (i.e., the complete discrete analogy of all continuous equations in every grid cell, even for nondiagonal conductivity tensors), a special conductivity averaging that does not require the grid to be small compared to layering or fractures, and a spectrally optimal grid refinement minimizing the error at the receiver locations and optimizing the approximation of the boundary conditions at infinity. All of these features significantly reduce the grid size and accelerate the computation of electromagnetic logs in 3D geometries without sacrificing accuracy.

194 citations


Book
01 Jan 2003
TL;DR: In this article, the authors present an algorithm for solving a Tridiagonal System of Linear Equations (TSOLE) problem with Dirichlet Boundary conditions, which is similar to the one we consider in this paper.
Abstract: (NOTE: Each chapter begins with An Overview.) 1. Getting Started. Algorithms. Convergence. Floating Point Numbers. Floating Point Arithmetic. 2. Rootfinding. Bisection Method. Method of False Position. Fixed Point Iteration. Newton's Method. The Secant Method and Muller's Method. Accelerating Convergence. Roots of Polynomials. 3. Systems of Equations. Gaussian Elimination. Pivoting Strategies. Norms. Error Estimates. LU Decomposition. Direct Factorization. Special Matrices. Iterative Techniques for Linear Systems: Basic Concepts and Methods. Iterative Techniques for Linear Systems: Conjugate-Gradient Method. Nonlinear Systems. 4. Eigenvalues and Eigenvectors. The Power Method. The Inverse Power Method. Deflation. Reduction to Tridiagonal Form. Eigenvalues of Tridiagonal and Hessenberg Matrices. 5. Interpolation and Curve Fitting. Lagrange Form of the Interpolating Polynomial. Neville's Algorithm. The Newton Form of the Interpolating Polynomial and Divided Differences. Optimal Interpolating Points. Piecewise Linear Interpolation. Hermite and Hermite Cubic Interpolation. Regression. 6. Numerical Differentiation and Integration. Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations. 7. Numerical Methods for Initial Value Problems of Ordinary Differential Equations. Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations. 8. Second-Order One-Dimensional Two-Point Boundary Value Problems. Finite Difference Method, Part I: The Linear Problem with Dirichlet Boundary Conditions. Finite Difference Method, Part II: The Linear Problem with Non-Dirichlet Boundary Conditions. Finite Difference Method, Part III: Nonlinear Problems. The Shooting Method, Part I: Linear Boundary Value Problems. The Shooting Method, Part II: Nonlinear Boundary Value Problems. 9. Finite Difference Method for Elliptic Partial Differential Equations. The Poisson Equation on a Rectangular Domain, I: Dirichlet Boundary Conditions. The Poisson Equation on a Rectangular Domain, II: Non-Dirichlet Boundary Conditions. Solving the Discrete Equations: Relaxation Schemes. Local Mode Analysis of Relaxation and the Multigrid Method. Irregular Domains. 10. Finite Difference Method for Parabolic Partial Differential Equations. The Heat Equation with Dirichlet Boundary Conditions. Stability. More General Parabolic Equations. Non-Dirichlet Boundary Conditions. Polar Coordinates. Problems in Two Space Dimensions. 11. Finite Difference Method for Hyperbolic Partial Differential Equations and the Convection-Diffusion Equation. Advection Equation, I: Upwind Differencing. Advection Equation, II: MacCormack Method. Convection-Diffusion Equation. The Wave Equation. Appendices. Appendix A. Important Theorems from Calculus. Appendix B. Algorithm for Solving a Tridiagonal System of Linear Equations. References. Index. Answers to Selected Problems.

Journal ArticleDOI
TL;DR: In this article, central and upwind compact schemes for spatial discretization have been analyzed with respect to accuracy in spectral space, numerical stability and dispersion relation preservation, and some well-known compact schemes that were found to be G-K-S and time stable are shown to be unstable for selective length scales by this analysis.

Journal ArticleDOI
TL;DR: In this paper, a high-order numerical method was proposed for large-eddy simulations, particularly those containing wall-bounded regions which are considered on stretched curvilinear meshes, where spatial derivatives are represented by a sixth-order compact approximation that is used in conjunction with a tenth-order non-dispersive filter.
Abstract: This work investigates a high-order numerical method which is suitable for performing large-eddy simulations, particularly those containing wall-bounded regions which are considered on stretched curvilinear meshes. Spatial derivatives are represented by a sixth-order compact approximation that is used in conjunction with a tenth-order non-dispersive filter. The scheme employs a time-implicit approximately factored finite-difference algorithm, and applies Newton-like subiterations to achieve second-order temporal and sixth-order spatial accuracy. Both the Smagorinsky and dynamic subgrid-scale stress models are incorporated in the computations, and are used for comparison along with simulations where no model is employed. Details of the method are summarized, and a series of classic validating computations are performed. These include the decay of compressible isotropic turbulence, turbulent channel flow, and the subsonic flow past a circular cylinder. For each of these cases, it was found that the method was robust and provided an accurate means of describing the flowfield, based upon comparisons with previous existing numerical results and experimental data. Published in 2003 by John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: This work considers the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations, and shows results displaying the sharpness of the estimates.
Abstract: We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of Δx only. For example, when polynomials of degree k are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order k+1/2 in the L2-norm, whereas the post-processed approximation is of order 2k + 1; if the exact solution is in L2 only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order k + 1/2 in L2(Ω0), where Ω0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

Journal ArticleDOI
TL;DR: The present method performs well in solving the two-dimensional Burgers' equations in fully implicit finite-difference form and is examined by comparison with other analytical and numerical results.

Journal ArticleDOI
TL;DR: In this article, a mathematical model to describe the pyrolysis of a single solid particle of biomass is developed by incorporating improvements in the existing model reported in literature, which couples the heat transfer equation with the chemical kinetics equations.

Journal ArticleDOI
TL;DR: An explicit finite-difference lattice Boltzmann method for curvilinear coordinates is obtained and is applied to a two-dimensional Poiseuille flow, an unsteady Couette flow, a lid-driven cavity flow, and a steady flow around a circular cylinder.
Abstract: In this paper a finite-difference-based lattice Boltzmann method for curvilinear coordinates is proposed in order to improve the computational efficiency and numerical stability of a recent method [R. Mei and W. Shyy, J. Comput. Phys. 143, 426 (1998)] in which the collision term of the Boltzmann Bhatnagar-Gross-Krook equation for discrete velocities is treated implicitly. In the present method, the implicitness of the numerical scheme is removed by introducing a distribution function different from that being used currently. As a result, an explicit finite-difference lattice Boltzmann method for curvilinear coordinates is obtained. The scheme is applied to a two-dimensional Poiseuille flow, an unsteady Couette flow, a lid-driven cavity flow, and a steady flow around a circular cylinder. The numerical results are in good agreement with the results of previous studies. Extensions to other lattice Boltzmann models based on nonuniform meshes are also discussed.

Journal ArticleDOI
TL;DR: In this article, a non-periodic Poisson-Boltzmann dynamics method with a nonperiodic boundary condition is proposed for simulation of biomolecules in dilute aqueous solutions.
Abstract: We have developed a well-behaved and efficient finite difference Poisson–Boltzmann dynamics method with a nonperiodic boundary condition. This is made possible, in part, by a rather fine grid spacing used for the finite difference treatment of the reaction field interaction. The stability is also made possible by a new dielectricmodel that is smooth both over time and over space, an important issue in the application of implicit solvents. In addition, the electrostatic focusing technique facilitates the use of an accurate yet efficient nonperiodic boundary condition: boundary grid potentials computed by the sum of potentials from individual grid charges. Finally, the particle–particle particle–mesh technique is adopted in the computation of the Coulombic interaction to balance accuracy and efficiency in simulations of large biomolecules. Preliminary testing shows that the nonperiodic Poisson–Boltzmann dynamics method is numerically stable in trajectories at least 4 ns long. The new model is also fairly efficient: it is comparable to that of the pairwise generalized Born solventmodel, making it a strong candidate for dynamics simulations of biomolecules in dilute aqueous solutions. Note that the current treatment of total electrostaticinteractions is with no cutoff, which is important for simulations of biomolecules. Rigorous treatment of the Debye–Huckel screening is also possible within the Poisson–Boltzmann framework: its importance is demonstrated by a simulation of a highly charged protein.

Journal ArticleDOI
TL;DR: In this paper, the steady-state, hydromagnetic forced convective boundary-layer flow of an incompressible Newtonian, electrically-conducting and heat-generating/absorbing fluid over a non-isothermal wedge in the presence of thermal radiation effects is considered.
Abstract: This work is focused on the steady-state, hydromagnetic forced convective boundary-layer flow of an incompressible Newtonian, electrically-conducting and heat-generating/absorbing fluid over a non-isothermal wedge in the presence of thermal radiation effects. The wedge surface is assumed permeable so as to allow for possible wall suction or injection. Also included in the model are the effects of viscous dissipation, Joule heating and stress work. The governing partial differential equations for this investigation are derived and transformed using a non-similarity transformation. In deriving the governing equations, a temperature-dependent heat source or sink term is employed and the Rossland approximation for the thermal radiation term is assumed to be valid. The obtained non-similar equations are solved numerically by an implicit, iterative, tri-diagonal finite-difference method. Comparisons with previously published work on various special cases of the problem are performed and the results are found to be in excellent agreement. Numerical results for the velocity and temperature profiles for a prescribed magnetic parameter as well as the development of the local skin-friction coefficient and local Nusselt number with the magnetic parameter are presented graphically and discussed. This is done in order to elucidate the influence of the various parameters involved in the problem on the solution.

Journal ArticleDOI
TL;DR: A systematic procedure based on nonlocal approximation is proposed for the construction of qualitatively stable nonstandard finite difference schemes for the logistic equation, the combustion model and the reaction-diffusion equation.

Journal ArticleDOI
TL;DR: In this article, a procedure is given that can easily assure the quality of numerical results by obtaining the residual at each point, which can be applied over general or irregular clouds of points.

Journal ArticleDOI
TL;DR: In this paper, a coupled system of two singularly perturbed linear reaction-diffusion two-point boundary value problems is examined, where the leading term is multiplied by a small positive parameter, but these parameters may have different magnitudes.
Abstract: A coupled system of two singularly perturbed linear reaction-diffusion two-point boundary value problems is examined. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solutions to the system have boundary layers that overlap and interact. The structure of these layers is analysed, and this leads to the construction of a piecewise-uniform mesh that is a variant of the usual Shishkin mesh. On this mesh central differencing is proved to be almost first-order accurate, uniformly in both small parameters. Supporting numerical results are presented for a test problem.

Journal ArticleDOI
TL;DR: This paper investigates the accurate numerical solution of the equations governing bed-load sediment transport and two approaches: a steady and an unsteady approach are discussed and five different formulations within these frameworks are derived.
Abstract: This paper investigates the accurate numerical solution of the equations governing bed-load sediment transport. Two approaches: a steady and an unsteady approach are discussed and five different formulations within these frameworks are derived. A flux-limited version of Roe's scheme is used with the different formulations on a channel test problem and the results compared.

Journal ArticleDOI
TL;DR: A numerical method for solving multi-scale parabolic problems based on the use of two different schemes for the original equation, at different grid level which allows to give numerical results at a much lower cost than solving the original equations.

Journal ArticleDOI
TL;DR: A class of numerical schemes for solving the HJB equation for stochastic control problems enters the framework of Markov chain approximations and generalizes the usual finite difference method, showing how to compute effectively the class of covariance matrices that is consistent with this set of points.
Abstract: We analyze a class of numerical schemes for solving the HJB equation for stochastic control problems, which enters the framework of Markov chain approximations and generalizes the usual finite difference method. The latter is known to be monotonic, and hence valid, only if the scaled covariance matrix is dominant diagonal. We generalize this result by, given the set of neighboring points allowed to enter the scheme, showing how to compute effectively the class of covariance matrices that is consistent with this set of points. We perform this computation for several cases in dimensions 2, 3, and 4.

Journal ArticleDOI
TL;DR: In this article, a pseudospectral method for solving the three-dimensional time-dependent Gross-Pitaevskii (GP) equation was proposed, and it was used to study the resonance dynamics of a trapped Bose-Einstein condensate induced by a periodic variation in the atomic scattering length.
Abstract: We suggest a pseudospectral method for solving the three-dimensional time-dependent Gross–Pitaevskii (GP) equation, and use it to study the resonance dynamics of a trapped Bose–Einstein condensate induced by a periodic variation in the atomic scattering length. When the frequency of oscillation of the scattering length is an even multiple of one of the trapping frequencies along the x, y or z direction, the corresponding size of the condensate executes resonant oscillation. Using the concept of the differentiation matrix, the partial-differential GP equation is reduced to a set of coupled ordinary differential equations, which is solved by a fourth-order adaptive step-size control Runge–Kutta method. The pseudospectral method is contrasted with the finite-difference method for the same problem, where the time evolution is performed by the Crank–Nicholson algorithm. The latter method is illustrated to be more suitable for a three-dimensional standing-wave optical-lattice trapping potential.

Journal ArticleDOI
01 Jan 2003
TL;DR: In this paper, a mathematical model to describe the detailed three-dimensional transient heat transfer process in friction stir welding (FSW) is presented, where the heat input from the tool shoulder is modelled as a frictional heat and the heat from a tool pin is modeled as a uniform volumetric heat generated by the plastic deformation near the pin.
Abstract: A mathematical model to describe the detailed three-dimensional transient heat transfer process in friction stir welding (FSW) is presented. This work is both theoretical and experimental. An explicit central differential scheme is used in solving the control equations, the heat transfer phenomena during the tool penetrating, the welding and the tool-removing periods that are studied dynamically. The heat input from the tool shoulder is modelled as a frictional heat and the heat from the tool pin is modelled as a uniform volumetric heat generated by the plastic deformation near the pin. The temperature variation during the welding is also measured to validate the calculated results. The calculated results are in good agreement with the experimental data.

Journal ArticleDOI
TL;DR: A nonlinear modal analysis approach based on the invariant manifold method proposed earlier by Boivin et al. as discussed by the authors is applied in this paperto perform the dynamic analysis of a micro switch, which is modeled as a clamped-clamped microbeam subjected to a transverseelectrostatic force.
Abstract: A nonlinear modal analysis approach based on the invariant manifoldmethod proposed earlier by Boivin et al. [10] is applied in this paperto perform the dynamic analysis of a micro switch. The micro switch ismodeled as a clamped-clamped microbeam subjected to a transverseelectrostatic force. Two kinds of nonlinearities are encountered in thenonlinear system: geometric nonlinearity of the microbeam associatedwith large deflection, and nonlinear coupling between two energydomains. Using Galerkin method, the nonlinear partial differentialgoverning equation is decoupled into a set of nonlinear ordinarydifferential equations. Based on the invariant manifold method, theassociated nonlinear modal shapes, and modal motion governing equationsare obtained. The equation of motion restricted to these manifolds,which provide the dynamics of the associated normal modes, are solved bythe approach of nonlinear normal forms. Nonlinearities and the pull-inphenomena are examined. The numerical results are compared with thoseobtained from the finite difference method. The estimate for the pull-involtage of the micro device is also presented.

Journal ArticleDOI
TL;DR: This paper introduces a fast numerical method for computing American option pricing problems governed by the Black--Scholes equation that is very efficient and gives better accuracy than the normal finite difference method.
Abstract: This paper introduces a fast numerical method for computing American option pricing problems governed by the Black--Scholes equation. The treatment of the free boundary is based on some properties of the solution of the Black--Scholes equation. An artificial boundary condition is also used at the other end of the domain. The finite difference method is used to solve the resulting problem. Computational results are given for some American call option problems. The results show that the new treatment is very efficient and gives better accuracy than the normal finite difference method.

Journal ArticleDOI
TL;DR: A model of transverse piano string vibration, second order in time, which models frequency-dependent loss and dispersion effects is presented here, and the waveguide model is extended to the case of several coupled strings.
Abstract: A model of transverse piano string vibration, second order in time, which models frequency-dependent loss and dispersion effects is presented here. This model has many desirable properties, in particular that it can be written as a well-posed initial-boundary value problem (permitting stable finite difference schemes) and that it may be directly related to a digital waveguide model, a digital filter-based algorithm which can be used for musical sound synthesis. Techniques for the extraction of model parameters from experimental data over the full range of the grand piano are discussed, as is the link between the model parameters and the filter responses in a digital waveguide. Simulations are performed. Finally, the waveguide model is extended to the case of several coupled strings.