Showing papers on "Finite difference published in 2009"
TL;DR: It is proved that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis and the stability is discussed using the Fourier method.
427 citations
TL;DR: In this article, the Laplace transform for the nabla derivative on the time scale of integers is introduced and properties of discrete fractional calculus in the sense of a backward difference are introduced and developed.
Abstract: Properties of discrete fractional calculus in the sense of a backward difference are introduced and developed. Exponential laws and a product rule are developed and relations to the forward fractional calculus are explored. Properties of the Laplace transform for the nabla derivative on the time scale of integers are developed and a fractional finite difference equation is solved with a transform method. As a corollary, two new identities for the gamma function are exhibited.
372 citations
TL;DR: It is shown that for the pressure treatment, an accurate Fourier representation can be used for more flexible boundary conditions than periodicity or free-slip, and this compromise fits particularly well for very high-resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments.
338 citations
TL;DR: This paper presents and compares two unconditionally energy stable finite-difference schemes for the phase field crystal equation and considers a new, fully second-order two-step algorithm that solves the nonlinear equations using an efficient nonlinear multigrid method.
287 citations
TL;DR: A robust multigrid method based on Gauss-Seidel smoothing is found to require special treatment of the boundary conditions along solid boundaries, and in particular on the sea bottom, and it is shown to provide convergent solutions over the full physical and discrete parameter space of interest.
238 citations
TL;DR: In this paper, the authors examined some practical numerical methods to solve a class of initial-boundary value problems for the fractional Fokker-Planck equation on a finite domain.
211 citations
TL;DR: In this article, a mimetic finite difference method for solving elliptic equations with tensor coefficients on polyhedral meshes was developed, and first-order convergence estimates in a mesh-dependent H 1 norm were derived.
Abstract: We developed a mimetic finite difference method for solving elliptic equations with tensor coefficients on polyhedral meshes. The first-order convergence estimates in a mesh-dependent H 1 norm are derived.
205 citations
TL;DR: In this article, a hybrid finite difference (HFD) method was used to solve a 2D bounce-averaged energy-pitch angle diffusion equation in the outer radiation belts.
Abstract: [1] In this study, we develop a new code which uses a hybrid finite difference (HFD) method to solve a 2-D bounce-averaged energy–pitch angle diffusion equation in the outer radiation belts. We implement the numerical algorithm by adopting a split operator technique, an implicit scheme for dealing with diagonal diffusion coefficients, and an alternative direction implicit scheme for off-diagonal (or cross) diffusion coefficients. We show that our code HFD can successfully overcome the unstable problem when the large and rapidly varying cross diffusion coefficients are included. Particularly, we examine whether resonant interactions with obliquely propagated whistler mode chorus and hiss waves result in a net acceleration or loss of relativistic electrons in the outer radiation belt L = 4.5. Numerical simulations show that chorus waves can yield significant acceleration of relativistic electrons, while hiss waves are primarily responsible for scattering equatorially mirroring electrons toward the loss cone.
190 citations
TL;DR: In this article, a modified numerical scheme for a class of Frac- tional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a fractional derivative (FD) is defined in the Riemann-Liouville sense is presented.
Abstract: This paper presents a modified numerical scheme for a class of Frac- tional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a Fractional Derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire time domain is divided into several sub- domains, and a fractional derivative (FDs) at a time node point is approx- imated using a modified Grunwald-Letnikov approach. For the first order derivative, the proposed modified Grunwald-Letnikov definition leads to a central difference scheme. When the approximations are substituted into the Fractional Optimal Control (FCO) equations, it leads to a set of alge- braic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system, and 2) as the sizes of the sub- domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.
181 citations
TL;DR: A new unified methodology to derive spatial finite-difference coefficients in the joint time-space domain to reduce numerical dispersion and can be easily extended to solve similar partial difference equations arising in other fields of science and engineering.
169 citations
TL;DR: First-order consistency, unconditional stability, and first-order convergence of the method are proven using a novel shifted version of the classical Grunwald finite difference approximation for the fractional derivatives.
TL;DR: In this paper, a two-dimensional mathematical model was theoretically developed to predict the temperature polarization profile of direct contact membrane distillation (DCMD) processes and a concurrent flat-plate device was designed to verify the theoretical prediction of pure water productivity on saline water desalination.
TL;DR: A stable and conservative high order multi-block method for the time-dependent compressible Navier-Stokes equations has been developed and stability and conservation are proved using summation-by-parts operators, weak interface conditions and the energy method.
TL;DR: This work derives explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection-diffusion equation and presents the order of accuracy of the schemes, and proves they are stable under certain conditions.
TL;DR: In this article, a hybrid scheme based on a set of 2DH extended Boussinesq equations for slowly varying bathymetries is introduced, which combines the finite volume technique, applied to solve the advective part of the equations, with the finite difference method, used to discretize dispersive and source terms.
TL;DR: The analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy and the numerical experiments show that the operators splitting methods have comparable discretization errors.
Abstract: We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.
TL;DR: In this paper, a numerical finite difference based formulation is proposed to effectively accommodate these two additional conditions, i.e., concrete cover repair or replacement and time dependent variation of the surface chloride ion concentration and diffusion coefficient.
Abstract: Service life of concrete structures under chloride environment can be predicted by formulations based on the mechanism of chloride ion diffusion. This mechanism can be mathematically described using the partial differential equation (PDE) of the Fick’s second law. One-dimensional PDE can be solved analytically by assuming constant surface chloride ion concentration and constant diffusion coefficient. However, the solution becomes more complicated when two additional conditions are included, i.e., concrete cover repair or replacement and time dependent variation of the surface chloride ion concentration and diffusion coefficient. In this paper, a numerical finite difference based formulation is proposed to effectively accommodate these two additional conditions. By virtue of numerical computation, the nonlinear initial chloride ion concentration can be treated in point-wise manner and both the time dependent surface chloride ion concentration and diffusion coefficient can be iteratively updated. Based on a Crank–Nicolson scheme within the finite difference method, a proper formulation accounting for space-dependent diffusion coefficient was derived; chloride ion concentration profiles are obtained and the service life of repaired concrete structures under chloride environment is predicted. Numerical examples and observations are finally presented.
TL;DR: In this article, a 2D finite-difference, frequency-domain method was developed for modeling viscoacoustic seismic waves in transversely isotropic media with a tilted symmetry axis.
Abstract: A 2D finite-difference, frequency-domain method was developed for modeling viscoacoustic seismic waves in transversely isotropic media with a tilted symmetry axis. The medium is parameterized by the P-wave velocity on the symmetry axis, the density, the attenuation factor, Thomsen’s anisotropic parameters δ and ϵ , and the tilt angle. The finite-difference discretization relies on a parsimonious mixed-grid approach that designs accurate yet spatially compact stencils. The system of linear equations resulting from discretizing the time-harmonic wave equation is solved with a parallel direct solver that computes monochromatic wavefields efficiently for many sources. Dispersion analysis shows that four grid points per P-wavelength provide sufficiently accurate solutions in homogeneous media. The absorbing boundary conditions are perfectly matched layers (PMLs). The kinematic and dynamic accuracy of the method wasassessed with several synthetic examples which illustrate the propagation of S-waves excited at t...
TL;DR: In this article, a high-order Boussinesq-type finite difference model was extended to simulate wave propagation out to a large number of times depth in a high order infinite difference model.
TL;DR: All the numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist.
Abstract: This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods. The main idea of the proposed vanishing moment method is to approximate a fully nonlinear second order PDE by a higher order, in particular, a quasilinear fourth order PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist.
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TL;DR: Block-to-block interface interpolation operators are constructed for several common high-order finite difference discretizations that maintain the strict stability, accuracy, and conservation properties of the base scheme even when nonconforming grids or dissimilar operators are used in adjoining blocks.
Abstract: Block-to-block interface interpolation operators are constructed for several common high-order finite difference discretizations. In contrast to conventional interpolation operators, these new interpolation operators maintain the strict stability, accuracy and conservation of the base scheme even when nonconforming grids or dissimilar operators are used in adjoining blocks. The stability properties of the new operators are verified using eigenvalue analysis, and the accuracy properties are verified using numerical simulations of the Euler equations in two spatial dimensions.
TL;DR: In this article, the feasibility of a hybrid scheme using Daubechies wavelet functions and the finite element method to obtain numerical solutions of some problems in structural mechanics is investigated.
TL;DR: Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation (“leap-frog” scheme) is used for the time discretization.
Abstract: In Grote et al. (SIAM J. Numer. Anal., 44:2408---2431, 2006) a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-dependent wave equation. In particular, optimal a-priori error bounds in the energy norm and the L 2-norm were derived for the semi-discrete formulation. Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation ("leap-frog" scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L 2-norm error over a finite time interval converges optimally as O(h p+1+Δt 2), where p denotes the polynomial degree, h the mesh size, and Δt the time step.
TL;DR: A new MFD method is presented for the Stokes problem on arbitrary polygonal meshes and its stability is analyzed, which allows the method to apply to a linear elasticity problem, as well.
TL;DR: Numerical solution of the Burgers’ equation is presented based on the cubic B-spline quasi-interpolation, by using the derivative of the quasi-Interpolation to approximate the spatial derivatives of the dependent variable and a low order forward difference to approximate the time derivative ofThe dependent variable.
TL;DR: In this article, the authors developed tools for numerical simulations of flame propagation with mesh-free radial basis functions (RBFs), which offer many distinct advantages over traditional finite difference, finite element, and finite volume methods.
Abstract: The purpose of this research was to develop tools for numerical simulations of flame propagation with mesh-free radial basis functions (RBFs). Mesh-free methods offer many distinct advantages over traditional finite difference, finite element, and finite volume methods. Traditional Lagrangian methods with significant swirl require mesh stiffeners and periodic remeshing to avoid excessive mesh distortion; such codes often require user interaction to repair the meshes before the simulation can proceed again. A propagating flame of infinite extent is simulated as a collection of normalized cells with periodic boundary conditions. Rather than capturing the flame front, it is tracked as a discontinuity. The flame front is approximated as a product of a Heaviside function in the normal propagation direction and a piece-wise continuous function represented by RBFs in the tangential direction. The cells are subdivided into the burned and unburned sub-domains approximated by two-dimensional periodic RBFs that are constrained to be strictly conservative. The underlying steady flow is vortical with an input turbulent intensity. The governing equations are rotationally and translationally transformed to produce exact differentials that are integrated exactly in time. In the present paper, the previous results of Aldredge who used a finite-difference level-set method were compared. The physical behavior was remarkably similar, whereas the finite-difference level-set method required 14 h of CPU time, the RBF approach required only 120 CPU seconds on a desktop computer for the case with the largest turbulent intensity. Although there are no other papers that tried to duplicate the original results of Aldredge, the results that are reported here are consistent with the physics observed in other experimental and numerical investigations.
TL;DR: A stable and accurate boundary treatment is derived for the second-order wave equation using narrow-diagonal summation by parts operators and the boundary conditions are imposed using a penalty method, leading to fully explicit time integration.
Abstract: A stable and accurate boundary treatment is derived for the second-order wave equation. The domain is discretized using narrow-diagonal summation by parts operators and the boundary conditions are imposed using a penalty method, leading to fully explicit time integration. This discretization yields a stable and efficient scheme. The analysis is verified by numerical simulations in one-dimension using high-order finite difference discretizations, and in three-dimensions using an unstructured finite volume discretization.
TL;DR: The improved high-order finite difference weighted essentially non-oscillatory WENO-Z method for solution of the hyperbolic conservation laws that govern the shocked carrier gas flow, and the ENO based algorithm is shown to be numerically stable and to accurately capture shocks, small flow features and particle dispersion.
01 Jan 2009
TL;DR: In this article, the authors present a systematical derivation of well models for other numerical methods such as standard finite element, control volume finite element and mixed finite element methods, which have particular applications to groundwater hydrology and petroleum reservoirs.
Abstract: Numerical simulation of fluid flow and transport processes in the subsurface must account for the presence of wells. The pressure at a gridblock that contains a well is different from the average pressure in that block and different from the flowing bottom hole pressure for the well [17]. Various finite difference well models have been developed to account for the difference. This paper presents a systematical derivation of well models for other numerical methods such as standard finite element, control volume finite element, and mixed finite element methods. Numerical results for a simple well example illustrating local grid refinement effects are given to validate these well models. The well models have particular applications to groundwater hydrology and petroleum reservoirs.
TL;DR: In this article, a three-dimensional Navier-Stokes solver for two-phase flow problems with surface tension is presented, where the free surface between the two fluid phases is tracked with a level set (LS) technique.
Abstract: In this paper we present a three-dimensional Navier–Stokes solver for incompressible two-phase flow problems with surface tension and apply the proposed scheme to the simulation of bubble and droplet deformation. One of the main concerns of this study is the impact of surface tension and its discretization on the overall convergence behavior and conservation properties.
Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin's projection approach. The free surface between the two fluid phases is tracked with a level set (LS) technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and a third-order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the LS function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second-order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface, which leads to a first-order convergence in the smoothing region.
We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature, and the application of our solver to the simulation of two rising bubble problems, one with small and one with large jumps in material parameters, and the simulation of a droplet deformation due to a shear flow in three space dimensions. Furthermore, we compare our three-dimensional results with those of quasi-two-dimensional and two-dimensional simulations. This comparison clearly shows the need for full three-dimensional simulations of droplet and bubble deformation to capture the correct physical behavior. Copyright © 2009 John Wiley & Sons, Ltd.