Showing papers on "Finite difference published in 2014"

New perspectives on polygonal and polyhedral finite element methods

Abstract: Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finite difference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary polygonal meshes was devised. The method was coined as the virtual element method (VEM), since it did not require the explicit construction of basis functions. This advance provides a more in-depth understanding of mimetic schemes, and also endows polygonal-based Galerkin methods with greater flexibility than three-node and four-node finite element methods. In the VEM, a projection operator is used to realize the decomposition of the stiffness matrix into two terms: a consistent matrix that is known, and a stability matrix that must be positive semi-definite and which is only required to scale like the consistent matrix. In this paper, we first present an overview of previous developments on conforming polygonal and polyhedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinate-based Galerkin method on polygonal and polyhedral elements. The consistent matrix of the VEM is adopted, and numerical quadrature with generalized barycentric coordinates is used to compute the stability matrix. This facilitates post-processing of field variables and visualization in the VEM, and on the other hand, provides a means to exactly satisfy the patch test with efficient numerical integration in polygonal and polyhedral finite elements. We present numerical examples that demonstrate the sound accuracy and performance of the proposed method. For Poisson problems in ℝ2 and ℝ3, we establish that linearly complete generalized barycentric interpolants deliver optimal rates of convergence in the L2-norm and the H1-seminorm.

Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach

Abstract: The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a nonlocal operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solutions with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.

The numerical solution of nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions

Abstract: In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers equation. Firstly, we obtain a time discrete scheme by approximating the first-order time derivative via forward finite difference formula, then we use Kansa's approach to approximate the spatial derivatives. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. Also, we show that convergence order of the time discrete scheme is O(@t). We solve the two-dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular, triangular and circular domains and also the three-dimensional case is solved on the cubical and spherical domains. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations. Also, several test problems including the three-dimensional case are given. Numerical examples confirm the efficiency of the proposed scheme.

Discrete systems behave as nonlocal structural elements: Bending, buckling and vibration analysis

Abstract: It is shown herein that the bending, buckling and vibration problems of a microstructured beam can be modeled by Eringen's nonlocal elasticity model. The microstructured model is composed of rigid periodic elements elastically connected by rotational springs. It is shown that this discrete system is the finite difference formulation of a continuous problem, i.e. the Euler-Bernoulli beam problem. Starting from the discrete equations, a continualization method leads to the formulation of an Eringen's type nonlocal equivalent continuum. The sensitivity phenomenon of the apparent nonlocal length scale with respect to the bending, the vibrations and the buckling analyses is investigated in more detail. A unified length scale can be used for the microstructured-based model with both nonlocal constitutive law and nonlocal governing equations. The Finite Difference Method is used for studying the exact discrete problem and leads to tractable engineering formula. The bending behaviour of the microstructured cantilever beam does not reveal any scale effect in the presence of concentrated loads. This scale invariance is not a deficiency of Eringen's nonlocality because it is in fact supported by the exact discreteness of the microstructured beam. A comparison of the discrete and the continuous problems (for both static and dynamics analyses) show the efficiency of the nonlocal-based modelling for capturing scale effects. As it has already been shown for buckling or vibrations studies, small scale effects tend to soften the material in this case.

Analysis of Compatible Discrete Operator schemes for elliptic problems on polyhedral meshes

Abstract: Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is assembled cellwise from local operators. We analyze two schemes depending on whether the potential degrees of freedom are attached to the vertices or to the cells of the primal mesh. We derive new functional analysis results on the discrete gradient that are the counterpart of the Sobolev embeddings. Then, we identify the two design properties of the local discrete Hodge operators yielding optimal discrete energy error estimates. Additionally, we show how these operators can be built from local nonconforming gradient reconstructions using a dual barycentric mesh. In this case, we also prove an optimal L 2 -error estimate for the potential for smooth solutions. Links with existing schemes (finite elements, finite volumes, mimetic finite differences) are discussed. Numerical results are presented on three-dimensional polyhedral meshes. The detailed material is available from http://hal.archives-ouvertes.fr/hal-00751284

Investment Lags - A Numerical Approach

Abstract: In this paper we use a mixture of numerical methods including finite difference and body fitted co-ordinates to form a robust stable numerical scheme to solve the investment lag model presented in the paper by Bar-Ilan and Strange (1996). This allows us to apply our methodology to models with different stochastic processes that does not have analytic solutions.

A Survey of High Order Schemes for the Shallow Water Equations

Abstract: In this paper, we survey our recent work on designing high order positivity- preserving well-balanced finite difference and finite volume WENO (weighted essen- tially non-oscillatory) schemes, and discontinuous Galerkin finite element schemes for solving the shallow water equations with a non-flat bottom topography. These schemes are genuinely high order accurate in smooth regions for general solutions, are essentially non-oscillatory for general solutions with discontinuities, and at the same time they preserve exactly the water at rest or the more general moving water steady state solutions. A simple positivity-preserving limiter, valid under suitable CFL condi- tion, has been introduced in one dimension and reformulated to two dimensions with triangular meshes, and we prove that the resulting schemes guarantee the positivity of the water depth.

Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables

Abstract: We solve the longstanding problem of defining a functional characterization of the spectrum of the transfer matrix associated with the most general spin-1/2 representations of the six-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to be equivalent to the known separation of variables (SOV) representation, hence proving that it defines a complete characterization of the transfer matrix spectrum. The polynomial form of the Q-function allows us to show that a finite system of generalized Bethe equations can also be used to describe the complete transfer matrix spectrum.

Decorrelation and phase-shift of coda waves induced by local changes: multiple scattering approach and numerical validation

Abstract: We report on theoretical predictions of the decorrelation and phase-shift of coda waves induced by local changes in multiple scattering media. Using the multiple scattering formalism, we show that both expressions (decorrelation and phase-shift) involve a same sensitivity kernel based on the intensity transport in the medium. We compare the kernels based on the diffusion approximation with the ones based on the radiative transfer approximation, showing that the latter is more accurate at short times or for changes located close to the source or the receiver. We also perform a series of numerical simulations of wave propagation (finite differences) to validate our models in different configurations.

Brownian dynamics without Green's functions.

Abstract: We develop a Fluctuating Immersed Boundary (FIB) method for performing Brownian dynamics simulations of confined particle suspensions. Unlike traditional methods which employ analytical Green's functions for Stokes flow in the confined geometry, the FIB method uses a fluctuating finite-volume Stokes solver to generate the action of the response functions “on the fly.” Importantly, we demonstrate that both the deterministic terms necessary to capture the hydrodynamic interactions among the suspended particles, as well as the stochastic terms necessary to generate the hydrodynamically correlated Brownian motion, can be generated by solving the steady Stokes equations numerically only once per time step. This is accomplished by including a stochastic contribution to the stress tensor in the fluid equations consistent with fluctuating hydrodynamics. We develop novel temporal integrators that account for the multiplicative nature of the noise in the equations of Brownian dynamics and the strong dependence of the mobility on the configuration for confined systems. Notably, we propose a random finite difference approach to approximating the stochastic drift proportional to the divergence of the configuration-dependent mobility matrix. Through comparisons with analytical and existing computational results, we numerically demonstrate the ability of the FIB method to accurately capture both the static (equilibrium) and dynamic properties of interacting particles in flow.

Seismic modeling by optimizing regularized staggered-grid finite-difference operators using a time-space-domain dispersion-relationship-preserving method

Abstract: The staggered-grid finite-difference (FD) method is widely used in numerical simulation of the wave equation. With stability conditions, grid dispersion often exists because of the discretization of the time and the spatial derivatives in thewave equation. Therefore, suppressing grid dispersion is a key problem for the staggered-grid FD schemes. To reduce the grid dispersion, the traditional method uses high-order staggered-grid schemes in the space domain. However, the wave is propagated in the time and space domain simultaneously. Therefore, some researchers proposed to derive staggered-grid FD schemes based on the time-space domain dispersion relationship. However, such methods were restricted to low frequencies and special angles of propagation. We have developed a regularizing technique to tackle the ill-conditioned property of the symmetric linear system and to stably provide approximate solutions of the FD coefficients for acoustic-wave equations. Dispersion analysis and seismic numerical simulations determined that the proposed method satisfies the dispersion relationship over a much wider range of frequencies and angles of propagation and can ensure FD coefficients being solved via a well-posed linear system and hence improve the forward modeling precision.

Nonlocal convection-diffusionvolume-constrained problems and jump processes

Abstract: We introduce the Cauchy and time-dependent volume-constrained problems associated with a linear nonlocal convection-diffusion equation. These problems are shown to be well-posed and correspond to conventional convection-diffusion equations as the region of nonlocality vanishes. The problems also share a number of features such as the maximum principle, conservation and dispersion relations, all of which are consistent with their corresponding local counterparts. Moreover, these problems are the master equations for a class of finite activity Levy-type processes with nonsymmetric Levy measure. Monte Carlo simulations and finite difference schemes are applied to these nonlocal problems, to show the effects of time, kernel, nonlocality and different volume-constraints.

A conservative finite difference scheme for Poisson---Nernst---Planck equations

Abstract: A macroscopic model to describe the dynamics of ion transport in ion channels is the Poisson---Nernst---Planck (PNP) equations. In this paper, we develop a finite-difference method for solving PNP equations, second-order accurate in both space and time. We use the physical parameters specifically suited toward the modeling of ion channels. We present a simple iterative scheme to solve the system of nonlinear equations resulting from discretizing the equations implicitly in time, which is demonstrated to converge in a few iterations. We place emphasis on ensuring numerical methods to have the same physical properties that the PNP equations themselves also possess, namely conservation of total ions, correct rates of energy dissipation, and positivity of the ion concentrations. We describe in detail an approach to derive a finite-difference method that preserves the total concentration of ions exactly in time. In addition, we find a set of sufficient conditions on the step sizes of the numerical method that assure positivity of the ion concentrations. Further, we illustrate that, using realistic values of the physical parameters, the conservation property is critical in obtaining correct numerical solutions over long time scales.

Addressing integration error for polygonal finite elements through polynomial projections: A patch test connection

Abstract: Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead to a deterioration of convergence of the finite element solutions. We propose a general remedy, inspired by techniques in the recent literature of mimetic finite differences, for restoring consistency and thereby ensuring the satisfaction of the patch test and recovering optimal rates of convergence. The proposed approach, based on polynomial projections of the basis functions, allows for the use of moderate number of integration points and brings the computational cost of polygonal finite elements closer to that of the commonly used linear triangles and bilinear quadrilaterals. Numerical studies of a two-dimensional scalar diffusion problem accompany the theoretical considerations.