Showing papers on "Finite difference method published in 2004"
TL;DR: In this paper, the authors developed practical numerical methods to solve one dimensional fractional advection-dispersion equations with variable coefficients on a finite domain and demonstrated the practical application of these results is illustrated by modeling a radial flow problem.
1,334 citations
TL;DR: A design criteria for a volume-of-fluid interface reconstruction algorithm to be second-order accurate is proposed, which is that it reproduce lines in two space dimensions or planes in three space dimensions exactly.
672 citations
TL;DR: This paper combines the forward time centered space (FTCS) method with the Grunwald--Letnikov discretization of the Riemann--Liouville derivative to obtain an explicit FTCS scheme for solving the fractional diffusion equation.
Abstract: A numerical method for solving the fractional diffusion equation, which could also be easily extended to other fractional partial differential equations, is considered. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald--Letnikov discretization of the Riemann--Liouville derivative to obtain an explicit FTCS scheme for solving the fractional diffusion equation. The stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well-known von Neumann method for nonfractional partial differential equations. The analytical stability bounds are in excellent agreement with numerical test. A comparison between exact analytical solutions and numerical predictions is made.
545 citations
TL;DR: This work presents results of 3D numerical simulations using a finite difference code featuring fixed mesh refinement (FMR), in which a subset of the computational domain is refined in space and time.
Abstract: We present results of 3D numerical simulations using a finite difference code featuring fixed mesh refinement (FMR), in which a subset of the computational domain is refined in space and time. We apply this code to a series of test cases including a robust stability test, a nonlinear gauge wave and an excised Schwarzschild black hole in an evolving gauge. We find that the mesh refinement results are comparable in accuracy, stability and convergence to unigrid simulations with the same effective resolution. At the same time, the use of FMR reduces the computational resources needed to obtain a given accuracy. Particular care must be taken at the interfaces between coarse and fine grids to avoid a loss of convergence at higher resolutions, and we introduce the use of 'buffer zones' as one resolution of this issue. We also introduce a new method for initial data generation, which enables higher order interpolation in time even from the initial time slice. This FMR system, 'Carpet', is a driver module in the freely available Cactus computational infrastructure, and is able to endow generic existing Cactus simulation modules ('thorns') with FMR with little or no extra effort.
525 citations
01 Jan 2004
489 citations
TL;DR: In this paper, a spectral non-iterative solution of the Euler-Lagrange equation is proposed for 3D active surface reconstruction of star-shaped surfaces parameterized in polar coordinates.
Abstract: Variational energy minimization techniques for surface reconstruction are implemented by evolving an active surface according to the solutions of a sequence of elliptic partial differential equations (PDE's). For these techniques, most current approaches to solving the elliptic PDE are iterative involving the implementation of costly finite element methods (FEM) or finite difference methods (FDM). The heavy computational cost of these methods makes practical application to 3D surface reconstruction burdensome. In this paper, we develop a fast spectral method which is applied to 3D active surface reconstruction of star-shaped surfaces parameterized in polar coordinates. For this parameterization the Euler-Lagrange equation is a Helmholtz-type PDE governing a diffusion on the unit sphere. After linearization, we implement a spectral non-iterative solution of the Helmholtz equation by representing the active surface as a double Fourier series over angles in spherical coordinates. We show how this approach can be extended to include region-based penalization. A number of 3D examples and simulation results are presented to illustrate the performance of our fast spectral active surface algorithms.
489 citations
TL;DR: A continuous normalized gradient flow (CNGF) is presented and its energy diminishing property is proved, which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of Bose--Einstein condensates (BEC).
Abstract: In this paper, we present a continuous normalized gradient flow (CNGF) and prove its energy diminishing property, which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of Bose--Einstein condensates (BEC). We also investigate the energy diminishing property for the discretization of the CNGF. Two numerical methods are proposed for such discretizations: one is the backward Euler centered finite difference (BEFD) method, the other is an explicit time-splitting sine-spectral (TSSP) method. Energy diminishing for BEFD and TSSP for the linear case and monotonicity for BEFD for both linear and nonlinear cases are proven. Comparison between the two methods and existing methods, e.g., Crank--Nicolson finite difference (CNFD) or forward Euler finite difference (FEFD), shows that BEFD and TSSP are much better in terms of preserving the energy diminishing property of the CNGF. Numerical results in one, two, and three dimensions with magnetic trap confinement potential, as well as a potential of a stirrer corresponding to a far-blue detuned Gaussian laser beam, are reported to demonstrate the effectiveness of BEFD and TSSP methods. Furthermore we observe that the CNGF and its BEFD discretization can also be applied directly to compute the first excited state solution in BEC when the initial data is chosen as an odd function.
413 citations
TL;DR: In this article, a finite difference operator approximating second derivatives and satisfying a summation by parts rule was derived for the fourth, sixth and eighth order case by using the symbolic mathematics software Maple.
394 citations
TL;DR: In this article, the equilibrium configurations of a vesicle membrane under elastic bending energy, with prescribed volume and surface area, were derived using a variational phase field method. But the authors only considered the axial symmetrical case.
378 citations
TL;DR: In this article, the heat and mass transfer characteristics of natural convection about a vertical surface embedded in a saturated porous medium subjected to a magnetic field is numerically studied, by taking into account the diffusion-thermo (Dufour) and thermal-diffusion (Soret) effects.
TL;DR: In this article, the authors compare different finite-difference schemes for two-dimensional (2D) acoustic frequency-domain forward modelling based on staggered-grid stencils.
Abstract: SUMMARY
We compare different finite-difference schemes for two-dimensional (2-D) acoustic frequency-domain forward modelling. The schemes are based on staggered-grid stencils of various accuracy and grid rotation strategies to discretize the derivatives of the wave equation. A combination of two staggered-grid stencils on the classical Cartesian coordinate system and the 45° rotated grid is the basis of the so-called mixed-grid stencil. This method is compared with a parsimonious staggered-grid method based on a fourth-order approximation of the first derivative operator. Averaging of the mass acceleration can be incorporated in the two stencils. Sponge-like perfectly matched layer absorbing boundary conditions are also examined for each stencil and shown to be effective.
The deduced numerical stencils are examined for both the wavelength content and azimuthal variation. The accuracy of the fourth-order staggered-grid stencil is slightly superior in terms of phase velocity dispersion to that of the mixed-grid stencil when averaging of the mass acceleration term is applied to the staggered-grid stencil.
For fourth-order derivative approximations, the classical staggered-grid geometry leads to a stencil that incorporates 13 grid nodes. The mixed-grid approach combines only nine grid nodes. In both cases, wavefield solutions are computed using a direct matrix solver based on an optimized multifrontal method. For this 2-D geometry, the staggered-grid strategy is significantly less efficient in terms of memory and CPU time requirements because of the enlarged bandwidth of the impedance matrix and increased number of coefficients in the discrete stencil.
Therefore, the mixed-grid approach should be suggested as the routine scheme for 2-D acoustic wave propagation modelling in the frequency domain.
TL;DR: In this paper, the authors describe the application of the rotated staggered-grid (RSG) finite-difference technique to the wave equations for anisotropic and viscoelastic media.
Abstract: We describe the application of the rotated staggered-grid (RSG) finite-difference technique to the wave equations for anisotropic and viscoelastic media. The RSG uses rotated finite-difference operators, leading to a distribution of modeling parameters in an elementary cell where all components of one physical property are located only at one single position. This can be advantageous for modeling wave propagation in anisotropic media or complex media, including high-contrast discontinuities, because no averaging of elastic moduli is needed. The RSG can be applied both to displacement-stress and to velocity-stress finite-difference (FD) schemes, whereby the latter are commonly used to model viscoelastic wave propagation. With a von Neumann-style anlysis, we estimate the dispersion error of the RSG scheme in general anisotropic media. In three different simulation examples, all based on previously published problems, we demonstrate the application and the accuracy of the proposed numerical approach.
TL;DR: In this paper, a fully non-linear finite difference model was developed based on inviscid flow equations for liquid sloshing induced by harmonic base excitations, which is valid for any water depth except for small depth when viscous effects would become important.
TL;DR: This work uses partial differential equation techniques to remove noise from digital images using a total-variation filter to smooth the normal vectors of the level curves of a noise image and finite difference schemes are used to solve these equations.
Abstract: In this work, we use partial differential equation techniques to remove noise from digital images. The removal is done in two steps. We first use a total-variation filter to smooth the normal vectors of the level curves of a noise image. After this, we try to find a surface to fit the smoothed normal vectors. For each of these two stages, the problem is reduced to a nonlinear partial differential equation. Finite difference schemes are used to solve these equations. A broad range of numerical examples are given in the paper.
TL;DR: It is demonstrated, using asymptotic expansions, that the error can be bounded very precisely indeed at the cost of few extra derivative evaluations, and in place of derivatives it is possible to use finite difference approximations, which renders the computation of error bounds even cheaper.
Abstract: The main theme of this paper is the construction of efficient, reliable and affordable error bounds for two families of quadrature methods for highly oscillatory integrals. We demonstrate, using asymptotic expansions, that the error can be bounded very precisely indeed at the cost of few extra derivative evaluations. Moreover, in place of derivatives it is possible to use finite difference approximations, with spacing inversely proportional to frequency. This renders the computation of error bounds even cheaper and, more importantly, leads to a new family of quadrature methods for highly oscillatory integrals that can attain arbitrarily high asymptotic order without computation of derivatives.
TL;DR: Numerical experiments show that the operator splitting methodology is much more efficient than the projected SOR, while the accuracy of both methods are similar.
TL;DR: The dispersion and loss in microstructured fibers are studied using a full-vectorial compact-2D finite-difference method in frequency-domain and a dielectric constant averaging technique using Ampere's law across the curved media interface is presented.
Abstract: The dispersion and loss in microstructured fibers are studied using a full-vectorial compact-2D finite-difference method in frequency-domain. This method solves a standard eigen-value problem from the Maxwell’s equations directly and obtains complex propagation constants of the modes using anisotropic perfectly matched layers. A dielectric constant averaging technique using Ampere’s law across the curved media interface is presented. Both the real and the imaginary parts of the complex propagation constant can be obtained with a high accuracy and fast convergence. Material loss, dispersion and spurious modes are also discussed.
TL;DR: This paper presents dissipation operators that preserve both stability and accuracy for high order finite difference approximations of initial boundary value problems.
Abstract: Stability for nonlinear convection problems using centered difference schemes require the addition of artificial dissipation. In this paper we present dissipation operators that preserve both stability and accuracy for high order finite difference approximations of initial boundary value problems.
TL;DR: Systematic numerical experiments indicate that a second order implicit time discretization of the viscous term, with the pressure and convective terms treated explicitly, is stable under the standard CFL condition.
Book•
25 Jun 2004
TL;DR: In this article, the authors present a two-step Exact Difference Scheme and its applications, as well as two-stage Difference Schemes Generated by Taylor's Decomposition.
Abstract: 1 Linear Difference Equations.- 1.1 Difference Equations of the First Order.- 1.2 Difference Equations of the Second Order.- 1.3 Difference Equations with Constant Coefficients.- 2 Difference Schemes for First-Order Differential Equations.- 2.1 Single-Step Exact Difference Scheme and Its Applications.- 2.2 Taylor's Decomposition on Two Points and Its Applications.- 3 Difference Schemes for Second-Order Differential Equations.- 3.1 Two-Step Exact Difference Scheme and Its Applications.- 3.2 Taylor's Decomposition on Three Points and Its Applications.- 4 Partial Differential Equations of Parabolic Type.- 4.1 A Cauchy Problem. Well-posedness.- 4.2 Difference Schemes Generated by an Exact Difference Scheme.- 4.3 Single-Step Difference Schemes Generated by Taylor's Decomposition.- 5 Partial Differential Equations of Elliptic Type.- 5.1 A Boundary-Value Problem. Well-posedness.- 5.2 Difference Schemes Generated by an Exact Difference Scheme.- 5.3 Two-Step Difference Schemes Generated by Taylor's Decomposition.- 6 Partial Differential Equations of Hyperbolic Type.- 6.1 A Cauchy Problem.- 6.2 Difference Schemes Generated by an Exact Difference Scheme.- 6.3 Two-Step Difference Schemes Generated by Taylor's Decomposition.- 7 Uniform Difference Schemes for Perturbation Problems.- 7.1 A Cauchy Problem for Parabolic Equations.- 7.2 A Boundary-Value Problem for Elliptic Equations.- 7.3 A Cauchy Problem for Hyperbolic Equations.- 8 Appendix: Delay Parabolic Differential Equations.- 8.1 The Initial-Value Differential Problem.- 8.2 The Difference Schemes.- Comments on the Literature.
TL;DR: Various numerical techniques will be developed and compared for solving the one-dimensional advection-diffusion equation with constant coefficient using the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett.
TL;DR: In this article, a constrained interpolation profile (CIP) method is adopted as the base scheme for the model and the wave-body interaction is treated as a multiphase problem, which has liquid (water), gas (air), and solid (wavemaker and floating body) phases.
Abstract: A CFD model is proposed for numerical simulations of extremely nonlinear free-surface flows such as wave impact phenomena and violent wave–body interactions. The constrained interpolation profile (CIP) method is adopted as the base scheme for the model. The wave–body interaction is treated as a multiphase problem, which has liquid (water), gas (air), and solid (wave-maker and floating body) phases. The flow is represented by one set of governing equations, which are solved numerically on a nonuniform, staggered Cartesian grid by a finite-difference method. The free surface as well as the body boundary are immersed in the computation domain and captured by different methods. In this article, the proposed numerical model is first described. Then to validate the accuracy and demonstrate the capability, several two-dimensional numerical simulations are presented, and compared with experiments and with computations by other numerical methods. The numerical results show that the present computation model is both robust and accurate for violent free-surface flows.
TL;DR: In this paper, a hybrid approach combining the conventional finite difference (FD) scheme and the mesh-free least square-based finite difference method (MLSFD) was proposed to simulate the two-dimensional steady and unsteady incompressible flows.
TL;DR: In this paper, a damage localization method based on changes in uniform load surface (ULS) curvature is developed for two-dimensional plate structures, which requires only the frequencies and mode shapes of the first few modes of the plate before and after damage.
TL;DR: A novel numerical method for a degenerate partial differential equation, called the Black-Scholes equation, governing option pricing, based on a fitted finite volume spatial discretization and an implicit time stepping technique is presented.
Abstract: In this paper we present a novel numerical method for a degenerate partial differential equation, called the Black-Scholes equation, governing option pricing. The method is based on a fitted finite volume spatial discretization and an implicit time stepping technique. To derive the error bounds for the spatial discretization of the method, we formulate it as a Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems defined on element edges. Stability of the discretization is proved and an error bound for the spatial discretization is established. It is also shown that the system matrix of the discretization is an M-matrix so that the discrete maximum principle is satisfied by the discretization. Numerical experiments are performed to demonstrate the effectiveness of the method.
TL;DR: In this article, the authors presented a modified BGK equation adapted to various flow regimes by the aid of the basic characteristics on molecular movement and collision approaching to equilibrium, and the discrete velocity ordinate method was developed and applied to the velocity distribution function to remove its continuous dependency on the velocity space.
TL;DR: In this article, the principle of lattice-Boltzmann techniques is recalled and some of the difficulties to simulate convective flows are discussed, and a hybrid scheme with lattice Boltzmann for fluid velocity variables and finite-difference for the temperature is proposed.
TL;DR: In this article, a method for simulating incompressible, imiscible, unsteady, Newtonian, multi-fluid flows with free surfaces is described, where a sharp interface separates fluids of different density and viscosity.
TL;DR: In this paper, a generalized reference model (Model I) incorporating all the above effects is proposed and further improved by proposing two simplified models (Models II and III) incorporating additional assumptions.