Showing papers on "Finite difference published in 2003"

Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing

Abstract: This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, we decompose a given (possible textured) image f into a sum of two functions u+v, where u∈BV is a function of bounded variation (a cartoon or sketchy approximation of f), while v is a function representing the texture or noise. To model v we use the space of oscillating functions introduced by Yves Meyer, which is in some sense the dual of the BV space. The new algorithm is very simple, making use of differential equations and is easily solved in practice. Finally, we implement the method by finite differences, and we present various numerical results on real textured images, showing the obtained decomposition u+v, but we also show how the method can be used for texture discrimination and texture segmentation.

Variation of the Electrophilicity Index along the Reaction Path.

Abstract: Some exact conditions for the extremals of the electrophilicity index, ω = μ(2)/2η (Parr, R. G.; von Szentpaly, L.; Liu, S. J. Am. Chem. Soc. 1999, 121, 1922), along an arbitrary reaction coordinate, have been carefully examined. Implications within the widely used finite difference approximation for the density-functional based reactivity descriptors, their relationship with the maximum hardness principle, and the reliability of the general relationships have been tested in the framework of computational evidence for some simple systems of chemical interest.

On using radial basis functions in a “finite difference mode” with applications to elasticity problems

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.

The effect of numerical errors and turbulence models in large-eddy simulations of channel flow, with and without explicit filtering

Abstract: Turbulent channel flow simulations are performed using second- and fourth-order finite difference codes. A systematic comparison of the large-eddy simulation (LES) results for different grid resolutions, finite difference schemes, and several turbulence closure models is performed. The use of explicit filtering to reduce numerical errors is compared to results from the traditional LES approach. Filter functions that are smooth in spectral space are used, as the findings of this investigation are intended for application of LES to complex domains. Explicit filtering introduces resolved subfilter-scale (RSFS) as well as subgrid-scale (SGS) turbulence terms. The former can be theoretically reconstructed; the latter must be modelled. The dynamic Smagorinsky model, the dynamic mixed model, and the new dynamic reconstruction model are all studied. It is found that for explicit filtering, increasing the reconstruction levels for the RSFS stress improves the mean velocity as well as the turbulence intensities. When compared to LES without explicit filtering, the difference in the mean velocity profiles is not large; however the turbulence intensities are improved for the explicit filtering case.

An Eulerian PDE approach for computing tissue thickness

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.

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

Abstract: We show how existing models for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments ("sedimentation with compression"' or "sedimentation-consolidation process"). For N solid particle species, this theory reduces in one space dimension to an $N\times N$ coupled system of quasi-linear degenerate convection-diffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N=3$ is illustrated by a numerical simulation obtained by the Kurganov-Tadmor central difference scheme for convection-diffusion problems. The numerical scheme exploits...

A Poisson–Boltzmann dynamics method with nonperiodic boundary condition

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.

On the implementation of perfectly matched layers in a three‐dimensional fourth‐order velocity‐stress finite difference scheme

Abstract: [1] Robust absorbing boundary conditions are central to the utility and advancement of 3-D numerical wave propagation methods. It is in general preferred that an absorbing boundary method be capable of broadband absorption, be efficient in terms of memory and computation time, and be widely stable in connection with sophisticated numerical schemes. Here we discuss these issues for a promising absorbing boundary method, perfectly matched layers (PML), as implemented in the widely used fourth-order accurate three-dimensional (3-D) staggered-grid velocity-stress finite difference (FD) scheme. Numerical results for point (explosive and double couple) and extended sources, velocity structures (homogeneous, 1-D and 3-D), and different thickness PML zones are excellent, in general, leaving no observable reflections in PML seismograms compared to the amplitudes of the primary phases. For both homogeneous half-space and 1-D models, typical amplitude reduction factors (with respect to the maximum trace amplitude) range between 1/100 and 1/625 for PML thicknesses of 5–20 nodes. A PML region of thickness 5 outperforms a simple exponential damping region of thickness 20 in a homogeneous half-space model by a factor of 3. We find that PML is effective across the simulation bandwidth. For example, permanent offset artifacts due to particularly poor absorption of long-period energy by the simple exponential damping are effectively absent when PML is used. The computational efficiency and storage requirements of PML, compared to the simple exponential damping, are reduced due to the need for only narrow absorbing regions. We also discuss stability and present the complete PML model for the 3-D velocity-stress system.

Hybrid finite-difference thermal lattice boltzmann equation

Abstract: We analyze the acoustic and thermal properties of athermal and thermal lattice Boltzmann equation (LBE) in 2D and show that the numerical instability in the thermal lattice Boltzmann equation (TLBE) is related to the algebraic coupling among different modes of the linearized evolution operator. We propose a hybrid finite-difference (FD) thermal lattice Boltzmann equation (TLBE). The hybrid FD-TLBE scheme is far superior over the existing thermal LBE schemes in terms of numerical stability. We point out that the lattice BGK equation is incompatible with the multiple relaxation time model.

Velocity-scalar filtered density function for large eddy simulation of turbulent flows

Abstract: A methodology termed the “velocity-scalar filtered density function” (VSFDF) is developed and implemented for large eddy simulation (LES) of turbulent flows. In this methodology, the effects of the unresolved subgrid scales (SGS) are taken into account by considering the joint probability density function (PDF) of the velocity and scalar fields. An exact transport equation is derived for the VSFDF in which the effects of the SGS convection and chemical reaction are closed. The unclosed terms in this equation are modeled in a fashion similar to that typically used in Reynolds-averaged simulation procedures. A system of stochastic differential equations (SDEs) which yields statistically equivalent results to the modeled VSFDF transport equation is constructed. These SDEs are solved numerically by a Lagrangian Monte Carlo procedure in which the Ito–Gikhman character of the SDEs is preserved. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed by comparison with results obtained by a finite difference LES procedure in which the corresponding transport equations for the first two SGS moments are solved. The VSFDF results are compared with those obtained by the Smagorinsky model, and all the results are assessed via comparison with data obtained by direct numerical simulation of a temporally developing mixing layer involving transport of a passive scalar. It is shown that the values of both the SGS and the resolved components of all second order moments including the scalar fluxes are predicted well by VSFDF. The sensitivity of the calculations to the model’s (empirical) constants are assessed and it is shown that the magnitudes of these constants are in the same range as those employed in PDF methods.

Ground states and dynamics of multi-component Bose-Einstein condensates

Abstract: We study numerically the time-independent vector Gross-Pitaevskii equations (VGPEs) for ground states and time-dependent VGPEs with (or without) an external driven field for dynamics describing a multi-component Bose-Einstein condensate (BEC) at zero or very low temperature. In preparation for the numerics, we scale the 3d VGPEs, approximately reduce it to lower dimensions, present a normalized gradient flow (NGF) to compute ground states of multi-component BEC, prove energy diminishing of the NGF which provides a mathematical justification, discretize it by the backward Euler finite difference (BEFD) which is monotone in linear and nonlinear cases and preserves energy diminishing property in linear case. Then we use a time-splitting sine-spectral method (TSSP) to discretize the time-dependent VGPEs with an external driven field for computing dynamics of multi-component BEC. The merit of the TSSP for VGPEs is that it is explicit, unconditionally stable, time reversible and time transverse invariant if the VGPEs is, `good' resolution in the semiclassical regime, of spectral order accuracy in space and second order accuracy in time, and conserves the total particle number in the discretized level. Extensive numerical examples in 3d for ground states and dynamics of multi-component BEC are presented to demonstrate the power of the numerical methods and to discuss the physics of multi-component Bose-Einstein condensates.

Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation

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.

An analysis of three different formulations of the discontinuous galerkin method for diffusion equations

Abstract: In this paper we present an analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. The first formulation yields an numerically inconsistent and weakly unstable scheme, while the other two formulations, the local discontinuous Galerkin approach and the Baumann–Oden approach, give stable and convergent results. When written as finite difference schemes, such a distinction among the three formulations cannot be easily analyzed by the usual truncation errors, because of the phenomena of supraconvergence and weak instability. We perform a Fourier type analysis and compare the results with numerical experiments. The results of the Fourier type analysis agree well with the numerical results.

High-order accurate methods in time-domain computational electromagnetics: A review

Abstract: Publisher Summary This chapter reviews the Maxwell's equations in the time domain and discusses boundary conditions, various simplifications, and standard normalizations. The chapter provides an overview of the classical phase-error analysis as a way of motivating the need to consider high-order accurate methods in time-domain electromagnetics, particularly as problems increase in size and complexity. The extensions of the Yee scheme and other more complex finite difference schemes are discussed. Higher-order schemes allow a significant reduction of the degrees of freedom with accuracy. For some applications it may be natural to consider the ultimate limit, leading to global or spectral methods. The chapter discusses the elements of spectral multidomain methods, which combine the accuracy of global methods with the geometric flexibility of a multielement formulation. The recent efforts on the development of high-order finite volume methods for the solution of Maxwell's equations are reviewed. The issues related to high-order time stepping and discrete stability are also discussed.

The simulation of piano string vibration: from physical models to finite difference schemes and digital waveguides.

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.