Showing papers on "Finite difference method published in 1995"

Implicit-explicit methods for time-dependent partial differential equations

Abstract: Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized partial differential equations (PDEs) of diffusion-convection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reaction-diffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes, and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods.For the prototype linear advection-diffusion equation, a stability analysis for first-, second-, third-, and fourth-order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified. Numerical experiments demonstrate that weak decay...

Finite‐difference time‐domain simulation of low‐frequency room acoustic problems

Abstract: This paper illustrates the use of a numerical time‐domain simulation based on the finite‐difference time‐domain (FDTD) approximation for studying low‐ and middle‐frequency room acoustic problems. As a direct time‐domain simulation, suitable for large modeling regions, the technique seems a good ‘‘brute force’’ approach for solving room acoustic problems. Some attention is paid in this paper to a few of the key problems involved in applying FDTD: frequency‐dependent boundary conditions, non‐Cartesian grids, and numerical error. Possible applications are illustrated with an example. An interesting approach lies in using the FDTD simulation to adapt a digital filter to represent the acoustical transfer function from source to observer, as accurately as possible. The approximate digital filter can be used for auralization experiments.

Order- N spectral method for electromagnetic waves

Abstract: We show that the eigenmodes for electromagnetic waves in an inhomogeneous dielectric medium can be obtained with an algorithm that scales linearly with the size of the system. The method employs discretization of the Maxwell equations in both the spatial and the time domain and the integration of the Maxwell equations in the time domain. The spectral intensity can then be obtained by a Fourier transform. We applied the method to a few problems of current interest, including the photonic band structure of a periodic dielectric structure, the effective dielectric constants of some three-dimensional and two-dimensional systems, and the defect states of a periodic dielectric structure with structural defects.

A compact fourth‐order finite difference scheme for the steady incompressible Navier‐Stokes equations

Abstract: SUMMARY We note in this study that the Navier-Stokes equations, when expressed in streamfunction-vorticity fonn, can be approximated to fourth--order accuracy with stencils extending only over a 3 x 3 square of points. The key advantage of the new compact fourth-order scheme is that it allows direct iteration for low~to-mediwn Reynolds numbers. Numerical solutions are obtained for the model problem of the driven cavity and compared with solutions available in the literature. For Re $1500 point-SOR iteration is used and the convergence is fast.

Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation

Abstract: In a previous work, the authors have presented a formalism for deriving systematically invariant, symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities. This formalism is herein cast in the context of exact finite difference calculus. The algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus, in the same sense as that of the corresponding differential equation being derivable from its associated energy function (a conserved quantity). A clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities, which are the consistent counterpart of the invariant quantities in the continuous case. Results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism, and that preserve energy or linear momentum, are discussed in the context of finite difference calculus. Some ...

Wave propagation in heterogeneous, porous media: A velocity‐stress, finite‐difference method

Abstract: A particle velocity-stress, finite-difference method is developed for the simulation of wave propagation in 2-D heterogeneous poroelastic media. Instead of the prevailing second-order differential equations, we consider a first-order hyperbolic system that is equivalent to Biot's equations. The vector of unknowns in this system consists of the solid and fluid particle velocity components, the solid stress components, and the fluid pressure. A MacCormack finite-difference scheme that is fourth-order accurate in space and second-order accurate in time forms the basis of the numerical solutions for Biot's hyperbolic system. An original analytic solution for a P-wave line source in a uniform poroelastic medium is derived for the purposes of source implementation and algorithm testing. In simulations with a two-layer model, additional «slow» compressional incident, transmitted, and reflected phases are recorded when the damping coefficient is small. This «slow» compressional wave is highly attenuated in porous media saturated by a viscous fluid. From the simulation we also verified that the attenuation mechanism introduced in Biot's theory is of secondary importance for «fast» compressional and rotational waves. The existence of seismically observable differences caused by the presence of pores has been examined through synthetic experiments that indicate that amplitude variation with offset may be observed on receivers and could be diagnostic of the matrix and fluid parameters. This method was applied in simulating seismic wave propagation over an expanded steam-heated zone in Cold Lake, alberta in an area of enhanced oil recovery (EOR) processing. The results indicate that a seismic surface survey can be used to monitor thermal fronts

Direct numerical simulation of controlled transition in a flat-plate boundary layer

Abstract: The three-dimensional development of controlled transition in a flat-plate boundary layer is investigated by direct numerical simulation (DNS) using the complete Navier-Stokes equations. The numerical investigations are based on the so-called spatial model, thus allowing realistic simulations of spatially developing transition phenomena as observed in laboratory experiments. For solving the Navier-Stokes equations, an efficient and accurate numerical method was developed employing fourth-order finite differences in the downstream and wall-normal directions and treating the spanwise direction pseudo-spectrally. The present paper focuses on direct simulations of the wind-tunnel experiments by Kachanov et al. (1984, 1985) of fundamental breakdown in controlled transition. The numerical results agreed very well with the experimental measurements up to the second spike stage, in spite of relatively coarse spanwise resolution. Detailed analysis of the numerical data allowed identification of the essential breakdown mechanisms. In particular, from our numerical data, we could identify the dominant shear layers and vortical structures that are associated with this breakdown process.

Flow Analysis of Injection Molds

Abstract: Part 1 The Current State of Simulation: Introduction, Stress and Strain in Fluid Mechanics, Material Properties of Polymers, Governing Equations, Approximations for Injection Molding, Numerical Methods for Solution Part 2 Improving Molding Simulation: Improved Fiber Orientation Modeling, Improved Mechanical Property Modeling, Long Fiber-Filled Materials, Crystallization, Effects of Crystallizations on Rheology and Thermal Properties, Colorant Effects, Prediction of Post-Molding Shrinkage and Warpage, Additional Issues of Injection-Molding Simulation, Epilogue Appendices: History of Injection-Molding Simulation, Tensor Notation, Derivation of Fiber Evolution Equations, Dimensional Analysis of Governing Equations, The Finite Difference Method, The Finite Element Method, Numerical Methods for the 2.5D Approximation, Three-Dimensional FEM for Mold Filling Analysis, Level Set Method, Full Form of Mori-Tanaka Model.

A new method for parameterization of phase shift and backscattering amplitude

Abstract: Parameterization of phase and backscattering amplitude with cubic splines is described. Using the cubic spline, the analytical partial derivatives of the plane wave EXAFS function can be calcalated. The use of analytical partial derivatives decreases the CPU time needed for a refinement by over 60% for a three shell system compared to a refinement with partial derivaties calculated with the finite difference method.

Propagation in linear dispersive media: finite difference time-domain methodologies

Abstract: Finite difference time-domain (FDTD) methodologies are presented for electromagnetic wave propagation in two different kinds of linear dispersive media: an Nth order Lorentz and an Mth order Debye medium. The temporal discretization is accomplished by invoking the central difference approximation for the temporal derivatives that appear in the first-order differential equations. From this, the final equations are temporally advanced using the classical leapfrog method. One-dimensional scattering from a dielectric slab is chosen for a test case. Provided that the maximum operating frequency times the time step is small and that the wave is adequately resolved in space, as shown in the error analysis, the agreement between the computed and exact solutions will be excellent. The attached data, which are associated with the four pole Lorentz dielectric and the five pole Debye medium, verify this assertion. >

A finite element method based on Whitney forms to solve Maxwell equations in the time domain

Abstract: A new finite element method in the time domain based on the Whitney forms is presented. Using edge elements and face elements for space discretization of the fields and a leap-frog scheme in time, the algorithm gives a direct way to solve Maxwell equations in general unstructured meshes. >

Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions

Abstract: We describe a finite difference solution technique for the full-vector waveguide equation based upon the alternating-direction-implicit (ADI) iterative method. Our technique accurately treats dielectric boundaries (including corners), requires minimal computer resources, and executes faster (by factors of 3-10) than other iterative approaches. In addition, we employ a transparent boundary condition that effectively removes the sensitivity of the calculated results to the size of the computational domain. This feature greatly facilitates the examination of modes near cutoff. >

Numerical solution of the sideways heat equation by difference approximation in time

Abstract: We consider a Cauchy problem for the heat equation in the quarter plane, where data are given at x=1 and a solution is sought in the interval 0

Density distribution of calcium-induced alginate gels. A numerical study

Abstract: Charged polysaccharides often form hydrogels in the presence of cations. In many applications the polymer network density distribution and associated physical properties are of major practical importance. Depending on the detailed conditions, the resulting gel density may vary from fully homogeneous to strongly inhomogeneous. We have established a simple set of coupled chemical reaction–diffusion equations to model the gelling process of calcium-induced alginate gels. The necessary algorithms for numerical solution of the resulting simultaneous parabolic differential equations have been developed both for one-dimensional models and three-ldimensional models with cylindrical or spherical symmetry. The algorithms make use of the Crank–Nicolson implicit finite difference method. The results of the numerical analyses of the gel formation can be divided into several different regimes depending on the physical and chemical parameters of the alginates and the cations. The numerical results are in good agreements with reported experimental results. © 1995 John Wiley & Sons, Inc.

Finite difference method for generalized Zakharov equations

Abstract: . A conservative difference scheme is presented for the initial-boundary value problem for generalized Zakharov equations. The scheme canbe implicit or semiexplicit depending on the choice of a parameter. On thebasis of a priori estimates and an inequality about norms, convergence of thedifference solution is proved in order 0(h2 + t2) , which is better than previous results. IntroductionThe Zakharov equations [20](1.1) iEt + Exx-NE = 0,(1.2) ^Ntt-{N+\E\2)xx = 0describe the propagation of Langmuir waves in plasmas. Here the complexunknown function E is the slowly varying envelope of the highly oscillatoryelectric field, and the unknown real function N denotes the fluctuation of the ion density about its equilibrium value.The global existence of a weak solution for the Zakharov equations in one dimension is proved in [19], and existence and uniqueness of a smooth solutionfor the equations are obtained provided smooth initial data are prescribed.Numerical methods for the Zakharov equations are studied only in [5, 9, 10,

Mathematical Modeling of Groundwater Pollution

Abstract: Contents: Introduction.- Hydrodynamic Dispersion in Porous Media.- Analytical Solutions of Hydrodynamic Dispersion Equations.- Finite Difference Methods and the Method of Characteristics for Hydrodynamic Dispersion Equations.- Finite Element Methods for Solving Hydrodynamic Dispersion Equations.- Numerical Solutions of Advection-Dominated Problems.- Mathematical Models of Groundwater Quality.- Applications of Groundwater Quality Models.- Conclusions.- Appendix A: The Related Parameters in the Modeling of Mass Transport in Porous Media.- Appendix B: A FORTRAN Program for Simultaneously Simulating Groundwater Flow and Quality.- References.

A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation

Abstract: The Cahn-Hilliard equation is a nonlinear evolutionary equation that is of fourth order in space. In this paper a linearized finite difference scheme is derived by the method of reduction of order. It is proved that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. The coefficient matrix of the difference system is symmetric and positive definite, so many well-known iterative methods (e.g. Gauss-Seidel, SOR) can be used to solve the system.

Acoustic streaming induced in focused Gaussian beams

Abstract: Axisymmetric flow equations for a viscous incompressible fluid are transformed into the vorticity transport and Poisson’s equations They are numerically solved via a finite difference method imposing appropriate initial and boundary conditions A model source of 1‐cm radius and 5‐cm focal length with Gaussian amplitude distribution radiates 5‐MHz ultrasound beams in water Numerical examples are shown for buildup of acoustic streaming along and across the acoustic axis Evidently, hydrodynamic nonlinearity has an essential effect on the streaming generation in comparison with a linear flow case; the nonlinearity reduces the streaming velocity in the focal and prefocal region, whereas it tends to accelerate the flow in the postfocal region

A comparison of numerical techniques for modeling electromagnetic dispersive media

Abstract: A comparison of various time domain numerical techniques to model material dispersion is presented. Methods that model the material dispersion via a convolution integral as well as those that use a differential equation representation are considered. We have shown how the convolution integral arising in the electromagnetic constitutive relation can be approximated by the trapezoidal rule of numerical integration and implemented using a newly derived one-time-step recursion relation. The superiority of the new method, in terms of accuracy and computer resources, over four previously published techniques is demonstrated on the problem of a transient electromagnetic plane wave propagating in a dispersive media. All of the methods considered are easily incorporated into 3-D codes where the requirement for efficiency is very important.