Showing papers on "Finite difference published in 2002"
TL;DR: A simple technique is adopted which ensures metric cancellation and thus ensures freestream preservation even on highly distorted curvilinear meshes, and metric cancellation is guaranteed regardless of the manner in which grid speeds are defined.
950 citations
TL;DR: An efficient and accurate semianalytical method to map the molecular surface of a molecule onto a three‐dimensional lattice and a procedure that calculates induced surface charges from the FDPB solutions and then uses these charges in the calculation of reaction field energies.
Abstract: This article describes a number of algorithms that are designed to improve both the efficiency and accuracy of finite difference solutions to the Poisson-Boltzmann equation (the FDPB method) and to extend its range of application. The algorithms are incorporated in the DelPhi program. The first algorithm involves an efficient and accurate semianalytical method to map the molecular surface of a molecule onto a three-dimensional lattice. This method constitutes a significant improvement over existing methods in terms of its combination of speed and accuracy. The DelPhi program has also been expanded to allow the definition of geometrical objects such as spheres, cylinders, cones, and parallelepipeds, which can be used to describe a system that may also include a standard atomic level depiction of molecules. Each object can have a different dielectric constant and a different surface or volume charge distribution. The improved definition of the surface leads to increased precision in the numerical solutions of the PB equation that are obtained. A further improvement in the precision of solvation energy calculations is obtained from a procedure that calculates induced surface charges from the FDPB solutions and then uses these charges in the calculation of reaction field energies. The program allows for finite difference grids of large dimension; currently a maximum of 571 3 can be used on molecules containing several thousand atoms and charges. As described elsewhere, DelPhi can also treat mixed salt systems containing mono- and divalent ions and provide electrostatic free energies as defined by the nonlinear PB equation.
682 citations
01 Nov 2002
TL;DR: An accurate and efficient numerical method to solve the coupled Cahn-Hilliard/Navier-Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity, and solves the Navier- Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank-Nicolson-based projection methods.
Abstract: Phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near-critical interfacial phenomena, phase separation under shear, and microstructure evolution during solidification. However, because interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations require resolution of very thin layers to capture the physics of the problems studied. This demands robust numerical methods that can efficiently achieve high resolution and accuracy, especially in three dimensions. We present here an accurate and efficient numerical method to solve the coupled Cahn-Hilliard/Navier-Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity. The numerical method is a time-split scheme that combines a novel semi-implicit discretization for the convective Cahn-Hilliard equation with an innovative application of high-resolution schemes employed for direct numerical simulations of turbulence. This new semi-implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the Cahn-Hilliard equation at the same cost as that of an explicit scheme. It is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity. Moreover, we solve the Navier-Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank-Nicolson-based projection methods. For channel geometries, the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction. The capabilities of the method are demonstrated with several examples including phase separation with, and without, shear in two and three dimensions. The method effectively resolves interfacial layers of as few as three mesh points. The numerical examples show agreement with analytical solutions and scaling laws, where available, and the 3D simulations, in the presence of shear, reveal rich and complex structures, including strings.
456 citations
TL;DR: A new mode solver is described which uses Yee's 2-D mesh and an index averaging technique to provide a full-vectorial finite-difference analysis of microstructured optical fibers.
Abstract: In this paper we present a full-vectorial finite-difference analysis of microstructured optical fibers. A new mode solver is described which uses Yee's 2-D mesh and an index averaging technique. The modal characteristics are calculated for both conventional optical fibers and microstructured optical fibers. Comparison with previous finite difference mode solvers and other numerical methods is made and excellent agreement is achieved.
450 citations
TL;DR: An efficient implementation of the finite difference Poisson–Boltzmann solvent model based on the Modified Incomplete Cholsky Conjugate Gradient algorithm, which gives rather impressive performance for both static and dynamic systems.
Abstract: We report here an efficient implementation of the finite difference Poisson–Boltzmann solvent model based on the Modified Incomplete Cholsky Conjugate Gradient algorithm, which gives rather impressive performance for both static and dynamic systems. This is achieved by implementing the algorithm with Eisenstat's two optimizations, utilizing the electrostatic update in simulations, and applying prudent approximations, including: relaxing the convergence criterion, not updating Poisson–Boltzmann-related forces every step, and using electrostatic focusing. It is also possible to markedly accelerate the supporting routines that are used to set up the calculations and to obtain energies and forces. The resulting finite difference Poisson–Boltzmann method delivers efficiency comparable to the distance-dependent dielectric model for a system tested, HIV Protease, making it a strong candidate for solution-phase molecular dynamics simulations. Further, the finite difference method includes all intrasolute electrostatic interactions, whereas the distance dependent dielectric calculations use a 15-A cutoff. The speed of our numerical finite difference method is comparable to that of the pair-wise Generalized Born approximation to the Poisson–Boltzmann method. © 2002 Wiley Periodicals, Inc. J Comput Chem 23: 1244–1253, 2002
429 citations
TL;DR: In this paper, a time-splitting spectral approximation for the Schrodinger equation in the semiclassical regime is proposed. But the authors consider the case where the Planck constant e is small and require the spatial mesh size h = O(e) and the time step k = o(e).
401 citations
TL;DR: An efficient finite difference model of blood flow through the coronary vessels is developed and applied to a geometric model of the largest six generations of the coronary arterial network by constraining the form of the velocity profile across the vessel radius.
Abstract: An efficient finite difference model of blood flow through the coronary vessels is developed and applied to a geometric model of the largest six generations of the coronary arterial network. By constraining the form of the velocity profile across the vessel radius, the three-dimensional Navier--Stokes equations are reduced to one-dimensional equations governing conservation of mass and momentum. These equations are coupled to a pressure-radius relationship characterizing the elasticity of the vessel wall to describe the transient blood flow through a vessel segment. The two step Lax--Wendroff finite difference method is used to numerically solve these equations. The flow through bifurcations, where three vessel segments join, is governed by the equations of conservation of mass and momentum. The solution to these simultaneous equations is calculated using the multidimensional Newton--Raphson method. Simulations of blood flow through a geometric model of the coronary network are presented demonstrating phy...
329 citations
TL;DR: It is shown that the full Navier?Stokes solver is between first- and second-order accurate and reproduces results from well-studied benchmark problems in viscous fluid flow and the robustness of the code on flow in a complex domain is demonstrated.
305 citations
TL;DR: A semi-implicit numerical model for the 3D Navier-Stokes equations on unstructured grids is derived and discussed in this article, where the governing differential equations are discretized by means of a finite difference-finite volume algorithm which is robust, very efficient, and applies to barotropic and baroclinic, hydrostatic and nonhydrostatic, and one-, two-, and three-dimensional flow problems.
226 citations
TL;DR: This paper presents a novel computational approach, the discrete singular convolution (DSC) algorithm, for analysing plate structures, and demonstrates that different methods of implementation for the present algorithm can be deduced from a single starting point.
Abstract: This paper presents a novel computational approach, the discrete singular convolution (DSC) algorithm, for analysing plate structures. The basic philosophy behind the DSC algorithm for the approximation of functions and their derivatives is studied. Approximations to the delta distribution are constructed as either bandlimited reproducing kernels or approximate reproducing kernels. Unified features of the DSC algorithm for solving differential equations are explored. It is demonstrated that different methods of implementation for the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. The use of the algorithm for the vibration analysis of plates with internal supports is discussed. Detailed formulation is given to the treatment of different plate boundary conditions, including simply supported, elastically supported and clamped edges. This work paves the way for applying the DSC approach in the following paper to plates with complex support conditions, which have not been fully addressed in the literature yet.
222 citations
TL;DR: In this paper, the theoretical basis and the numerical implementation of free-vortex filament methods are reviewed for application to the prediction and analysis of helicopter rotor wakes, with a discussion of finite difference approximations to these equations and various numerical solution techniques.
Abstract: The theoretical basis and the numerical implementation of free-vortex filament methods are reviewed for application to the prediction and analysis of helicopter rotor wakes. The governing equations for the problem are described, with a discussion of finite difference approximations to these equations and various numerical solution techniques. Both relaxation and time-marching wake solution techniques are reviewed. It is emphasized how the careful consideration of stability and convergence (grid-independent behavior) are important to ensure a physically correct wake solution. The implementation of viscous diffusion and filament straining effects are also discussed. The need for boundary condition corrections to compensate for the inevitable wake truncation are described. Algorithms to accelerate the wake solution using velocity field interpolation are shown to reduce computational costs without a loss of accuracy. Several challenging examples of the application of free-vortex filament methods to helicopter rotor problems are shown, including multirotor configurations, flight near the ground, maneuvering flight conditions, and descending flight through the vortex ring state
TL;DR: General results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi- Bellman equations with variable coecients are obtained using systematically a tricky idea of N.V. Krylov.
Abstract: Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.
TL;DR: In this paper, a nonlinear liquid sloshing inside a partially filled rectangular tank has been investigated, where the fluid is assumed to be homogeneous, isotropic, viscous, Newtonian and exhibit only limited compressibility.
TL;DR: An unconditionally stable explicit pseudodynamic algorithm is proposed herein that can be used to perform pseudodynamic tests without using any iterative scheme or extra hardware that is generally needed by the currently available implicit pseudodynamic algorithms.
Abstract: An unconditionally stable explicit pseudodynamic algorithm is proposed herein. This pseudodynamic algorithm can be implemented as simply as the very commonly used explicit pseudodynamic algorithms, such as the central difference method and the Newmark explicit method as reported in 1959. Thus, it can be used to perform pseudodynamic tests without using any iterative scheme or extra hardware that is generally needed by the currently available implicit pseudodynamic algorithms. This integration method is second-order accurate and the most promising property of this explicit pseudodynamic algorithm is its unconditional stability. In addition, it possesses much better error propagation properties when compared to the Newmark explicit method and the central difference method.
TL;DR: In this paper, a hybrid numerical method for modeling the evolution of sharp phase interfaces on fixed grids is presented, where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front, and enrichment strategies of the eXtended Finite Element Method (X-FEM) are employed to represent the jump in temperature gradient that governs the velocity of the phase boundary.
Abstract: A hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented. We focus attention on two-dimensional solidification problems, where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front. The enrichment strategies of the eXtended Finite Element Method (X-FEM) are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary. A new approach with the X-FEM is suggested for this class of problems whereby the partition of unity is constructed with C1(Ω) polynomials and enriched with a C0(Ω) function. This approach leads to jumps in temperature gradient occurring only at the phase boundary, and is shown to significantly improve estimates for the front velocity. Temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment. In conjunction with a finer finite difference grid, the Level Set method is used to represent the evolution of the phase interface. An iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary. The robustness and utility of the method is demonstrated with several benchmark problems of phase transformation. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: A general approach to construct second and third order accurate, fully discrete (in both space and time) entropy conservative schemes for weak solutions containing nonclassical regularization-sensitive shock waves.
Abstract: We consider weak solutions of (hyperbolic or hyperbolic-elliptic) systems of conservation laws in one-space dimension and their approximation by finite difference schemes in conservative form. The systems under consideration are endowed with an entropy-entropy flux pair. We introduce a general approach to construct second and third order accurate, fully discrete (in both space and time) entropy conservative schemes. In general, these schemes are fully nonlinear implicit, but in some important cases can be explicit or linear implicit. Furthermore, semidiscrete entropy conservative schemes of arbitrary order are presented. The entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization-sensitive shock waves. Finally, specific examples are investigated and tested numerically. Our approach extends the results and techniques by Tadmor [in Numerical Methods for Compressible Flows---Finite Difference, Element and Volume Techniques, ASME, New York, 1986, pp. 149--158], LeFloch and Rohde [SIAM J. Numer. Anal., 37 (2000), pp. 2023--2060].
TL;DR: In this article, the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials is reviewed and extended for nonorthogonal, nonsmooth, structured and unstructured computational grids.
Abstract: This paper reviews and extends the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials. These difference operators satisfy the fundamental identities, conservation laws and theorems of vector and tensor calculus on nonorthogonal, nonsmooth, structured and unstructured computational grids. We provide explicit approximations for equations in two dimensions with discontinuous anisotropic diffusion tensors. We mention the similarities and differences between the new methods and mixed finite element or hybrid mixed finite element methods.
TL;DR: A special representation of the noise is considered, and it is compared with general representations of noises in the infinite dimensional setting and the effects of the noises on the accuracy of the approximations are illustrated.
Abstract: This paper is concerned with the numerical approximation of some linear stochastic partial differential equations with additive noises. A special representation of the noise is considered, and it is compared with general representations of noises in the infinite dimensional setting. Convergence analysis and error estimates are presented for the numerical solution based on the standard finite difference and finite element methods. The effects of the noises on the accuracy of the approximations are illustrated. Results of the numerical experiments are provided.
TL;DR: In this paper, the authors applied the rotated staggered grid to simulate the propagation of elastic waves in a 2D or 3D medium containing cracks, pores or free surfaces without hard-coded boundary conditions.
Abstract: The modelling of elastic waves in fractured media with an explicit finite-difference scheme causes instability problems on a staggered grid when the medium possesses high-contrast discontinuities (strong heterogeneities). For the present study we apply the rotated staggered grid. Using this modified grid it is possible to simulate the propagation of elastic waves in a 2D or 3D medium containing cracks, pores or free surfaces without hard-coded boundary conditions. Therefore it allows an efficient and precise numerical study of effective velocities in fractured structures. We model the propagation of plane waves through a set of different, randomly cracked media. In these numerical experiments we vary the wavelength of the plane waves, the crack porosity and the crack density. The synthetic results are compared with several static theories that predict the effective P- and S-wave velocities in fractured materials in the long wavelength limit. For randomly distributed and randomly orientated, rectilinear, non-intersecting, thin, dry cracks, the numerical simulations of velocities of P-, SV- and SH-waves are in excellent agreement with the results of the modified (or differential) self-consistent theory. On the other hand for intersecting cracks, the critical crack-density (porosity) concept must be taken into account. To describe the wave velocities in media with intersecting cracks, we propose introducing the critical crack-density concept into the modified self-consistent theory. Numerical simulations show that this new formulation predicts effective elastic properties accurately for such a case.
TL;DR: In this article, a numerical method capable of simulating viscoelastic free surface flow of an Oldroyd-B fluid was developed for the computation of the non-Newtonian extra-stress components on rigid boundaries.
Abstract: This work is concerned with the development of a numerical method capable of simulating viscoelastic free surface flow of an Oldroyd-B fluid. The basic equations governing the flow of an Oldroyd-B fluid are considered. A novel formulation is developed for the computation of the non-Newtonian extra-stress components on rigid boundaries. The full free surface stress conditions are employed. The resulting governing equations are solved by a finite difference method on a staggered grid, influenced by the ideas of the marker-and-cell (MAC) method. Numerical results demonstrating the capabilities of this new technique are presented for a number of problems involving unsteady free surface flows.
TL;DR: In this paper, the authors present an efficient algorithm for solving nonlinear heat problems involving phase change. But the authors do not consider the nonlinearity of the phase change problem in this paper.
TL;DR: This paper solves an unconstrained minimization problem on the entire space of functions, using the projection on the sphere of any arbitrary function, and shows how this formulation can be used in practice, for problems with both isotropic and anisotropic diffusion.
Abstract: We propose in this paper an alternative approach for computing p-harmonic maps and flows: instead of solving a constrained minimization problem on SN-1, we solve an unconstrained minimization problem on the entire space of functions. This is possible, using the projection on the sphere of any arbitrary function. Then we show how this formulation can be used in practice, for problems with both isotropic and anisotropic diffusion, with applications to image processing, using a new finite difference scheme.
TL;DR: In this paper, a 3D finite-difference solution is implemented for simulating induction log responses in the quasi-static limit that include the wellbore and bedding that exhibits transverse anisotropy.
Abstract: A 3-D finite-difference solution is implemented for simulating induction log responses in the quasi-static limit that include the wellbore and bedding that exhibits transverse anisotropy. The finite-difference code uses a staggered grid to approximate a vector equation for the electric field. The resulting linear system of equations is solved to a predetermined error level using iterative Krylov subspace methods. To accelerate the solution at low induction numbers (LINs), a new preconditioner is developed. This new preconditioner splits the electric field into curl-free and divergence-free projections, which allows for the construction of an approximate inverse operator. Test examples show up to an order of magnitude increase in speed compared to a simple Jacobi preconditioner. Comparisons with analytical and mode matching solutions demonstrate the accuracy of the algorithm.
TL;DR: In this paper, the authors present a computational method for quasi 3D unsteady flows of thin liquid films on a solid substrate, which includes surface tension as well as gravity forces in order to model realistically the spreading on an arbitrarily inclined substrate.
TL;DR: In this paper, a lattice Boltzmann model was proposed for 2D advection and anisotropic dispersion equation (AADE) based on the Bhatnagar, Gross and Krook (BGK) model.
TL;DR: A Lax--Wendroff time discretization procedure for high order finite difference weighted essentially nonoscillatory schemes to solve hyperbolic conservation laws and is more cost effective than the Runge--KuttaTime discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics.
Abstract: In this paper we develop a Lax--Wendroff time discretization procedure for high order finite difference weighted essentially nonoscillatory schemes to solve hyperbolic conservation laws. This is an alternative method for time discretization to the popular TVD Runge--Kutta time discretizations. We explore the possibility in avoiding the local characteristic decompositions or even the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining nonoscillatory properties for problems with strong shocks. As a result, the Lax--Wendroff time discretization procedure is more cost effective than the Runge--Kutta time discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics.
TL;DR: This paper presents a general approach for the stability analysis of the time-domain finite-element method (TDFEM) for electromagnetic simulations, which determines the stability by analyzing the root-locus map of a characteristic equation and evaluating the spectral radius of the finite element system matrix.
Abstract: This paper presents a general approach for the stability analysis of the time-domain finite-element method (TDFEM) for electromagnetic simulations. Derived from the discrete system analysis, the approach determines the stability by analyzing the root-locus map of a characteristic equation and evaluating the spectral radius of the finite element system matrix. The approach is applicable to the TDFEM simulation involving dispersive media and to various temporal discretization schemes such as the central difference, forward difference, backward difference, and Newmark methods. It is shown that the stability of the TDFEM is determined by the material property and by the temporal and spatial discretization schemes. The proposed approach is applied to a variety of TDFEM schemes, which include: (1) time-domain finite-element modeling of dispersive media; (2) time-domain finite element-boundary integral method; (3) higher order TDFEM; and (4) orthogonal TDFEM. Numerical results demonstrate the validity of the proposed approach for stability analysis.
TL;DR: In this paper, the discretization error of semilinear stochastic evolution equations in Lp-spaces is investigated, and the implicit Euler, the explicit Euler scheme and the Crank-Nicholson scheme are considered.
TL;DR: By extending the latter approach, this work performs a complete analysis of convergence of the TGM under the sole assumption that f is nonnegative and with a zero at $x^0=0$ of finite order.
Abstract: Summary. The solution of large Toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers [13, 14, 22]. The technique was modified in [6, 36] and a rigorous proof of convergence of the TGM (two-grid method) was given in the special case where the generating function has only a zero at x 0 =0 of order at most two. Here, by extending the latter approach, we perform a complete analysis of convergence of the TGM under the sole assumption that f is nonnegative and with a zero at x 0 =0 of finite order. An extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic PDEs with nonconstant coefficients and of any order is then discussed.
TL;DR: In this paper, a 3D finite-element scheme for direct current resistivity modeling is presented, where the singularity is removed by formulating the problem in terms of the secondary potential, which improves the accuracy considerably.
Abstract: SUMMARY A 3-D finite-element scheme for direct current resistivity modelling is presented. The singularity is removed by formulating the problem in terms of the secondary potential, which improves the accuracy considerably. The resulting system of linear equations is solved using the conjugate gradient method. The incomplete Cholesky preconditioner with a scaled matrix has been proved to be faster than the symmetric successive overrelaxation preconditioner. A compact storage scheme fully utilizes the sparsity and symmetry of the system matrix. The finite-element (FE) and a previously developed finite-difference (FD) scheme are compared in detail. Generally, both schemes show good agreement, the relative error in apparent resistivity for a vertical dike model presented in this paper is less than 0.5 per cent overall. The FD scheme produces larger errors near the conductivity contrast, whereas the FE scheme requires approximately 3.4 times as much storage as the FD scheme and is less robust with respect to coarse grids. As an improvement to the forward modelling scheme, a modified singularity removal technique is presented. A horizontally layered earth or a vertical contact is regarded as the normal structure, the solution of which is the primary potential. The effect of this technique is demonstrated by two examples: a cube in two-layered earth and a cube near a vertical contact.