Showing papers on "Finite difference method published in 2007"

3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study

Abstract: We present a finite-difference frequency-domain method for 3D visco-acoustic wave propagation modeling. In the frequency domain, the underlying numerical problem is the resolution of a large sparse system of linear equations whose right-hand side term is the source. This system is solved with a massively parallel direct solver. We first present an optimal 3D finite-difference stencil for frequency-domain modeling. The method is based on a parsimonious staggered-grid method. Differential operators are discretized with second-order accurate staggered-grid stencils on different rotated coordinate systems to mitigate numerical anisotropy. An antilumped mass strategy is implemented to minimize numerical dispersion. The stencil incorporates 27 grid points and spans two grid intervals. Dispersion analysis shows that four grid points per wavelength provide accurate simulations in the 3D domain. To assess the feasibility of the method for frequency-domain full-waveform inversion, we computed simulations in the 3D SEG/EAGE overthrust model for frequencies 5, 7, and 10 Hz. Results confirm the huge memory requirement of the factorization (several hundred Figabytes) but also the CPU efficiency of the resolution phase (few seconds per shot). Heuristic scalability analysis suggests that the memory complexity of the factorization is O(35N(4)) for a N-3 grid. Our method may provide a suitable tool to perform frequency-domain full-waveform inversion using a large distributed-memory platform. Further investigation is still necessary to assess more quantitatively the respective merits and drawbacks of time- and frequency-domain modeling of wave propagation to perform 3D full-waveform inversion.

The finite-difference time-domain method for modeling of seismic wave propagation

Abstract: We present a review of the recent development in finite-difference time-domain modeling of seismic wave propagation and earthquake motion. The finite-difference method is a robust numerical method applicable to structurally complex media. Due to its relative accuracy and computational efficiency it is the dominant method in modeling earthquake motion and it also is becoming increasingly more important in the seismic industry and for structural modeling. We first introduce basic formulations and properties of the finite-difference schemes including promising recent advances. Then we address important topics as material discontinuities, realistic attenuation, anisotropy, the planar free surface boundary condition, free-surface topography, wavefield excitation (including earthquake source dynamics), non-reflecting boundaries, and memory optimization and parallelization.

MultiStencils Fast Marching Methods: A Highly Accurate Solution to the Eikonal Equation on Cartesian Domains

Abstract: A wide range of computer vision applications require an accurate solution of a particular Hamilton-Jacobi (HJ) equation known as the Eikonal equation. In this paper, we propose an improved version of the fast marching method (FMM) that is highly accurate for both 2D and 3D Cartesian domains. The new method is called multistencils fast marching (MSFM), which computes the solution at each grid point by solving the Eikonal equation along several stencils and then picks the solution that satisfies the upwind condition. The stencils are centered at each grid point and cover its entire nearest neighbors. In 2D space, two stencils cover 8-neighbors of the point, whereas in 3D space, six stencils cover its 26-neighbors. For those stencils that are not aligned with the natural coordinate system, the Eikonal equation is derived using directional derivatives and then solved using higher order finite difference schemes. The accuracy of the proposed method over the state-of-the-art FMM-based techniques has been demonstrated through comprehensive numerical experiments.

A finite volume numerical approach for coastal ocean circulation studies: Comparisons with finite difference models

Abstract: [1] An unstructured grid, finite volume, three-dimensional (3-D) primitive equation coastal ocean model (FVCOM) has been developed for the study of coastal ocean and estuarine circulation by Chen et al. (2003a). The finite volume method used in this model combines the advantage of finite element methods for geometric flexibility and finite difference methods for simple discrete computation. Currents, temperature, and salinity are computed using an integral form of the equations, which provides a better representation of the conservative laws for mass, momentum, and heat. Detailed comparisons are presented here of FVCOM simulations with analytical solutions and numerical simulations made with two popular finite difference models (the Princeton Ocean Model and Estuarine and Coastal Ocean Model (ECOM-si)) for the following idealized cases: wind-induced long-surface gravity waves in a circular lake, tidal resonance in rectangular and sector channels, freshwater discharge onto the continental shelf with curved and straight coastlines, and the thermal bottom boundary layer over the slope with steep bottom topography. With a better fit to the curvature of the coastline using unstructured nonoverlapping triangle grid cells, FVCOM provides improved numerical accuracy and correctly captures the physics of tide-, wind-, and buoyancy-induced waves and flows in the coastal ocean. This model is suitable for applications to estuaries, continental shelves, and regional basins that feature complex coastlines and bathymetry.

Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations

Abstract: . We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton-Jacobi-Bellman equations by introducing a new notion of consistency. Our results are robust and general - they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank-Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.

A Semi-Lagrangian Approach for Natural Gas Storage Valuation and Optimal Operation

Abstract: The valuation of a gas storage facility is characterized as a stochastic control problem, resulting in a Hamilton-Jacobi-Bellman (HJB) equation. In this paper, we present a semi-Lagrangian method for solving the HJB equation for a typical gas storage valuation problem. The method is able to handle a wide class of spot price models that exhibit mean-reverting seasonality dynamics and price jumps. We develop fully implicit and Crank-Nicolson timestepping schemes based on a semi-Lagrangian approach and prove the convergence of fully implicit timestepping to the viscosity solution of a modified HJB equation posed on a bounded domain, provided that a strong comparison result holds. The semi-Lagrangian approach avoids Policy-type iterations required by an implicit finite difference method without requiring additional cost. Numerical experiments are presented for several variants of the basic scheme.

A stable finite difference method for the elastic wave equation on complex geometries with free surfaces

Abstract: A stable and explicit second order accurate finite difference method for the elastic wave equation in curvilinear coordinates is presented. The discretization of the spatial operators in the method is shown to be self-adjoint for free-surface, Dirichlet and periodic boundary conditions. The fully discrete version of the method conserves a discrete energy to machine precision.

Numerical simulation of flows around two circular cylinders by mesh‐free least square‐based finite difference methods

Abstract: In this paper, the mesh-free least square-based finite difference (MLSFD) method is applied to numerically study the flow field around two circular cylinders arranged in side-by-side and tandem configurations. For each configuration, various geometrical arrangements are considered, in order to reveal the different flow regimes characterized by the gap between the two cylinders. In this work, the flow simulations are carried out in the low Reynolds number range, that is, Re = 100 and 200. Instantaneous vorticity contours and streamlines around the two cylinders are used as the visualization aids. Some flow parameters such as Strouhal number, drag and lift coefficients calculated from the solution are provided and quantitatively compared with those provided by other researchers.

Direct Numerical Simulations of Flow Past an Array of Distributed Roughness Elements

Abstract: Direct numerical simulation was used to describe the subsonic flow past an array of distributed cylindrical roughness elements mounted on a flat plate. Solutions were obtained for element heights corresponding to a roughness-based Reynolds number (Re k ) of both 202 and 334. The numerical method used a sixth-order-accurate centered compact finite difference scheme to represent spatial derivatives, which was used in conjunction with a tenth-order low-pass Pade-type nondispersive filter operator to maintain stability. An implicit approximately factored time-marching algorithm was employed, and Newton-like subiterations were applied to achieve second-order temporal accuracy. Calculations were carried out on a massively parallel computing platform, using domain decomposition to distribute subzones on individual processors. A high-order overset grid approach preserved spatial accuracy on the mesh system used to represent the roughness elements. Features of the flowfields are described, and results of the computations are compared with experimentally measured velocity components of the time-mean flowfield, which are available only for Re k = 202. Flow about the elements is characterized by a system of two weak corotating horseshoe vortices. For Re k = 334, an unstable shear layer emanating from the top of the cylindrical element generated nonlinear unsteady disturbances of sufficient amplitude to produce explosive bypass transition downstream of the array. The Re k = 202 case displayed exponential growth of turbulence energy in the streamwise direction, which may eventually result in transition.

Computational methods for large-scale 3D acoustic finite-difference modeling: A tutorial

Abstract: We present a set of methods for modeling wavefields in three dimensions with the acoustic-wave equation. The primary applications of these modeling methods are the study of acquisition design, multiple suppression, and subsalt imaging for surface-streamer and ocean-bottom recording geometries. We show how to model the acoustic wave equation in three dimensions using limited computer memory, typically using a single workstation, leading to run times on the order of a few CPU hours to a CPU day. The structure of the out-of-core method presented is also used to improve the efficiency of in-core modeling, where memory-to-cache-to-memory data flow is essentially the same as the data flow for an out-of-core method. Starting from the elastic-wave equation, we develop a vector-acoustic algorithm capable of efficiently modeling multicomponent data in an acoustic medium. We show that data from this vector-acoustic algorithm can be used to test upgoing/downgoing separation of P-waves recorded by ocean-bottom seismic...

Simulation of rapidly varying flow using an efficient TVD–MacCormack scheme

Abstract: SUMMARY An efficient numerical scheme is outlined for solving the SWEs (shallow water equations) in environmental flow; this scheme includes the addition of a five-point symmetric total variation diminishing (TVD) term to the corrector step of the standard MacCormack scheme. The paper shows that the discretization of the conservative and non-conservative forms of the SWEs leads to the same finite difference scheme when the source term is discretized in a certain way. The non-conservative form is used in the solution outlined herein, since this formulation is simpler and more efficient. The time step is determined adaptively, based on the maximum instantaneous Courant number across the domain. The bed friction is included either explicitly or implicitly in the computational algorithm according to the local water depth. The wetting and drying process is simulated in a manner which complements the use of operator-splitting and two-stage numerical schemes. The numerical model was then applied to a hypothetical dam-break scenario, an experimental dam-break case and an extreme flooding event over the Toce River valley physical model. The predicted results are free of spurious oscillations for both sub- and super-critical flows, and the predictions compare favourably with the experimental measurements. Copyright q 2006 John Wiley & Sons, Ltd.

On the accuracy of finite-difference solutions for nonlinear water waves

Abstract: This paper considers the relative accuracy and efficiency of low- and high-order finite-difference discretisations of the exact potential-flow problem for nonlinear water waves. The method developed is an extension of that employed by Li and Fleming (Coastal Engng 30: 235–238, 1997) to allow arbitrary-order finite-difference schemes and a variable grid spacing. Time-integration is performed using a fourth-order Runge–Kutta scheme. The linear accuracy, stability and convergence properties of the method are analysed and high-order schemes with a stretched vertical grid are found to be advantageous relative to second-order schemes on an even grid. Comparison with highly accurate periodic solutions shows that these conclusions carry over to nonlinear problems and that the advantages of high-order schemes improve with both increasing nonlinearity and increasing accuracy tolerance. The combination of non-uniform grid spacing in the vertical and fourth-order schemes is suggested as optimal for engineering purposes.

Discrete Orthogonal Decomposition and Variational Fluid Flow Estimation

Abstract: We exploit the mimetic finite difference method introduced by Hyman and Shashkov to present a framework for estimating vector fields and related scalar fields (divergence, curl) of physical interest from image sequences. Our approach provides a basis for consistent definitions of higher-order differential operators, for the analysis and a novel stability result concerning second-order div-curl regularizers, for novel variational schemes to the estimation of solenoidal (divergence-free) image flows, and to convergent numerical methods in terms of subspace corrections.

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

Abstract: A methodology termed the “velocity-scalar filtered mass density function” (VSFMDF) is developed and implemented for large eddy simulation (LES) of variable-density turbulent reacting flows. This methodology is based on the extension of the previously developed “velocity-scalar filtered density function” method for constant-density flows. In the VSFMDF, the effects of the unresolved subgrid scales (SGS) are taken into account by considering the joint probability density function of the velocity and scalar fields. An exact transport equation is derived for the VSFMDF in which the effects of SGS convection and chemical reaction are in closed forms. The unclosed terms in this equation are modeled in a fashion similar to that in Reynolds-averaged simulation procedures. A set of stochastic differential equations (SDEs) are considered which yield statistically equivalent results to the modeled VSFMDF transport equation. The SDEs are solved numerically by a Lagrangian Monte Carlo procedure in which the Ito-Gikhma...

Fast finite-difference time-domain modeling for marine-subsurface electromagnetic problems

Abstract: In the low-frequency limit, the displacement currents in the Maxwell equations can be neglected. However, for numerical simulations, a small displacement current should be present to achieve numerical stability. This requirement leads to a large range of propagation velocities with high velocities for the high frequencies and low velocities for the low frequencies. As a consequence, the number of time steps may become large. I show that it is possible to transform mathematically the original physical problem to one that has propagation velocities with less frequency dependence. Hence, the number of time steps necessary for a signal to travel a certain distance with the lowest velocity is significantly reduced. A typical example shows a reduction in computational time by a factor of 40. A comparison of the solutions from plane-layered modeling in the frequency and wavenumber domain and the proposed method shows good agreement between the two. The proposed method can also be used for other systems of diffusive equations.

Water Entry and Exit of a Horizontal Circular Cylinder

Abstract: In this paper we describe the fully nonlinear free-surface deformations of initially calm water caused by the water entry and water exit of a horizontal circular cylinder with both forced and free vertical motions. Two-dimensional flow conditions are assumed in the study. This has relevance for marine operations as well as for the ability to predict large amplitude motions of floating sea structures. A new numerical method called the CIP (Constrained Interpolation Profile) method is used to solve the problem. In this paper, the circular cylinder and free surface interaction is treated as a multiphase problem, which has liquid (water), gas (air), and solid (circular cylinder) 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 is immersed in the computational domain. The numerical results of the water entry and exit force, the free surface deformation and the vertical motion of the cylinder are compared with experimental results, and favorable agreement is obtained.