Showing papers on "Finite difference published in 2001"

Review Article Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

Abstract: In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobilike equations.

High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD

Abstract: In recent years high order numerical methods have been widely used in computational fluid dynamics (CFD), to effectively resolve complex flow features using meshes which are reasonable for today''s computers. In this paper we review and compare three types of high order methods being used in CFD, namely the weighted essentially non-oscillatory (WENO) finite difference methods, the WENO finite volume methods, and the discontinuous Galerkin (DG) finite element methods. We summarize the main features of these methods, from a practical user''s point of view, indicate their applicability and relative strength, and show a few selected numerical examples to demonstrate their performance on illustrative model CFD problems.

Modeling multiphase non-isothermal fluid flow and reactive geochemical transport in variably saturated fractured rocks: 1. methodology

Abstract: Reactive fluid flow and geochemical transport in unsaturated fractured rocks have received increasing attention for studies of contaminant transport, ground- water quality, waste disposal, acid mine drainage remediation, mineral deposits, sedimentary diagenesis, and fluid-rock interactions in hydrothermal systems. This paper presents methods for modeling geochemical systems that emphasize: (1) involvement of the gas phase in addition to liquid and solid phases in fluid flow, mass transport, and chemical reactions; (2) treatment of physically and chemically heterogeneous and fractured rocks, (3) the effect of heat on fluid flow and reaction properties and processes, and (4) the kinetics of fluid-rock interaction. The physical and chemical process model is embodied in a system of partial differential equations for flow and transport, coupled to algebraic equations and ordinary differential equations for chemical interactions. For numerical solution, the continuum equations are discretized in space and time. Space discretization is based on a flexible integral finite difference approach that can use irregular gridding to model geologic structure; time is discretized fully implicitly as a first-order finite difference. Heterogeneous and fractured media are treated with a general multiple interacting continua method that includes double-porosity, dual-permeability, and multi-region models as special cases. A sequential iteration approach is used to treat the coupling between fluid flow and mass transport on the one hand, chemical reactions on the other. Applications of the methods developed here to variably saturated geochemical systems are presented in a companion paper (part 2, this issue).

Inequalities on Time Scales: A Survey

Abstract: The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics which is currently receiving considerable attention. Although the basic aim of this is to unify the study of differential and difference equations, it also extends these classical cases to cases “in between”. In this paper we present time scales versions of the inequalities: Holder, Cauchy-Schwarz, Minkowski, Jensen, Gronwall, Bernoulli, Bihari, Opial, Wirtinger, and Lyapunov. 1. Unifying Continuous and Discrete Analysis In 1988, Stefan Hilger [13] introduced the calculus on time scales which unifies continuous and discrete analysis. A time scale is a closed subset of the real numbers. We denote a time scale by the symbol T . For functions y defined on T , we introduce a so-called delta derivative y∆ . This delta derivative is equal to y (the usual derivative) if T = R is the set of all real numbers, and it is equal to ∆y (the usual forward difference) if T = Z is the set of all integers. Then we study dynamic equations f (t; y; y∆; y∆ 2 ; : : : ; y∆ n ) = 0; which may involve higher order derivatives as indicated. Along with such dynamic equations we consider initial values and boundary conditions. We remark that these dynamic equations are differential equations when T = R and difference equations when T = Z . Other kinds of equations are covered by them as well, such as q difference equations, where T = q := fqkj k 2 Zg[ f0g for some q > 1 and difference equations with constant step size, where T = hZ := fhkj k 2 Zg for some h > 0: Particularly useful for the discretization purpose are time scales of the form T = ftkj k 2 Zg where tk 2 R; tk < tk+1 for all k 2 Z: Mathematics subject classification (2000): 34A40, 39A13.

Sensitivity Analysis for Navier-Stokes Equations on Unstructured Meshes Using Complex Variables

Abstract: The use of complex variables for determining sensitivity derivatives for turbulent flows is examined. Although a step size parameter is required, the numerical derivatives are not subject to subtractive cancellation errors and, therefore, exhibit true second-order accuracy as the step size is reduced. As a result, this technique guarantees two additional digits of accuracy each time the step size is reduced one order of magnitude. This behavior is in contrast to the use of finite differences, which suffer from inaccuracies due to subtractive cancellation errors. In addition, the complex-variable procedure is easily implemented into existing codes

Contributions to the mathematics of the nonstandard finite difference method and applications

Abstract: We formalize the transfer of essential properties of the solution of a differential equation to the solution of a discrete scheme as qualitative stability with respect to the properties. This permits us to motivate some rules (viz. on the order of the difference equation, on the renormalization of the denominator of the discrete derivative, and on nonlocal approximation of nonlinear terms) used in the design of nonstandard finite difference schemes. Extensions of some models are considered, and numerical examples confirming the efficiency of the nonstandard approach are provided. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 518–543, 2001

Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients

Abstract: New finite difference methods using Cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients, singular sources, and nonsmooth or even discontinuous solutions. The new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques. The methods are shown to converge under certain conditions using comparison functions. The coefficient matrix of the resulting linear system of equations is an M-matrix and is coupled with a multigrid solver. Numerical examples are also provided to show the efficiency of the proposed methods.

Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations

Abstract: We consider the numerical solution of the stochastic partial differential equation ∂u/∂t = ∂ 2 u/∂x 2 + σ(u)W(x,t), where W is space-time white noise, using finite differences. For this equation Gyongy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals of W over a rectangular grid. We investigate the extent to which this order of convergence can be improved, and find that better approximations are possible for the case of additive noise (σ(u) = 1) if we wish to estimate space averages of the solution rather than pointwise estimates, or if we are permitted to generate other functionals of the noise. But for multiplicative noise (σ(u) = u) we show that no such improvements are possible.

A k-space method for large-scale models of wave propagation in tissue

Abstract: Large-scale simulation of ultrasonic pulse propagation in inhomogeneous tissue is important for the study of ultrasound-tissue interaction as well as for development of new imaging methods. Typical scales of interest span hundreds of wavelengths. This paper presents a simplified derivation of the k-space method for a medium of variable sound speed and density; the derivation clearly shows the relationship of this k-space method to both past k-space methods and pseudospectral methods. In the present method, the spatial differential equations are solved by a simple Fourier transform method, and temporal iteration is performed using a k-t space propagator. The temporal iteration procedure is shown to be exact for homogeneous media, unconditionally stable for "slow" (c(x)/spl les/c/sub 0/) media, and highly accurate for general weakly scattering media. The applicability of the k-space method to large-scale soft tissue modeling is shown by simulating two-dimensional propagation of an incident plane wave through several tissue-mimicking cylinders as well as a model chest wall cross section. A three-dimensional implementation of the k-space method is also employed for the example problem of propagation through a tissue-mimicking sphere. Numerical results indicate that the k-space method is accurate for large-scale soft tissue computations with much greater efficiency than that of an analogous leapfrog pseudospectral method or a 2-4 finite difference time-domain method. However, numerical results also indicate that the k-space method is less accurate than the finite-difference method for a high contrast scatterer with bone-like properties, although qualitative results can still be obtained by the k-space method with high efficiency. Possible extensions to the method, including representation of absorption effects, absorbing boundary conditions, elastic-wave propagation, and acoustic nonlinearity, are discussed.

3‐D electromagnetic anisotropy modeling using finite differences

Abstract: Electric anisotropy is considered an important property of hydrocarbon reservoirs. Its occurrence has great influence on estimation of formation water saturation and other properties derived from electromagnetic (EM) measurements. Conventional tools using coaxial coils often underestimate formation resistivity and thus overestimate water saturation. Multicomponent EM sensors provide the additional information needed for better resistivity‐based formation evaluation. We have developed a finite‐difference method to simulate multicomponent EM tools in a 3‐D anisotropic formation. The new method can model inhomogeneous media with arbitrary anisotropy. By using the coupled Maxwell’s equations, our method consumes about the same computational time to model an anisotropic formation as it would take to model an otherwise isotropic formation. We have verified the finite‐difference method using layered‐earth models that are typically encountered in hydrocarbon exploration and development. Our results show that the ...

Least-squares image resizing using finite differences

Abstract: We present an optimal spline-based algorithm for the enlargement or reduction of digital images with arbitrary (noninteger) scaling factors. This projection-based approach can be realized thanks to a new finite difference method that allows the computation of inner products with analysis functions that are B-splines of any degree n. A noteworthy property of the algorithm is that the computational complexity per pixel does not depend on the scaling factor a. For a given choice of basis functions, the results of our method are consistently better than those of the standard interpolation procedure; the present scheme achieves a reduction of artifacts such as aliasing and blocking and a significant improvement of the signal-to-noise ratio. The method can be generalized to include other classes of piecewise polynomial functions, expressed as linear combinations of B-splines and their derivatives.

Symmetry without symmetry: numerical simulation of axisymmetric systems using cartesian grids

Abstract: We present a new technique for the numerical simulation of axisymmetric systems. This technique avoids the coordinate singularities which often arise when cylindrical or polar-spherical coordinate finite difference grids are used, particularly in simulating tensor partial differential equations like those of 3+1 numerical relativity. For a system axisymmetric about the z axis, the basic idea is to use a three-dimensional Cartesian(x,y,z) coordinate grid which covers (say) the y=0 plane, but is only one finite-difference-molecule–width thick in the y direction. The field variables in the central y=0 grid plane can be updated using normal (x,y,z)-coordinate finite differencing, while those in the y≠ 0 grid planes can be computed from those in the central plane by using the axisymmetry assumption and interpolation. We demonstrate the effectiveness of the approach on a set of fully nonlinear test computations in 3+1 numerical general relativity, involving both black holes and collapsing gravitational waves.

Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design.

Abstract: We have designed high-efficiency finite-aperture diffractive optical elements (DOE's) with features on the order of or smaller than the wave-length of the incident illumination. The use of scalar diffraction theory is generally not considered valid for the design of DOE's with such features. However, we have found several cases in which the use of a scalar-based design is, in fact, quite accurate. We also present a modified scalar-based iterative design method that incorporates the angular spectrum approach to design diffractive optical elements that operate in the near-field and have sub-wavelength features. We call this design method the iterative angular spectrum approach (IASA). Upon comparison with a rigorous electromag-netic analysis technique, specifically, the finite difference time-domain method (FDTD), we find that our scalar-based design method is surprisingly valid for DOE's having sub-wavelength features.

A Finite Difference Interpretation of the Lattice Boltzmann Method

Abstract: Compared to conventional techniques in computational fluid dynamics, the lattice Boltzmann method (LBM) seems to be a completely different approach to solve the incompressible Navier-Stokes equations. The aim of this article is to correct this impression by showing the close relation of LBM to two standard methods: relaxation schemes and explicit finite difference discretizations. As a side effect, new starting points for a discretization of the incompressible Navier-Stokes equations are obtained.

Discrete transparent boundary conditions for the Schrödinger equation

Abstract: This paper is concerned with transparent boundary conditions for the one dimensional time–dependent Schrodinger equation. They are used to restrict the original PDE problem that is posed on an unbounded domain onto a finite interval in order to make this problem feasible for numerical simulations. The main focus of this article is on the appropriate discretization of such transparent boundary conditions in conjunction with some chosen discretization of the PDE (usually Crank–Nicolson finite differences in the case of the Schrodinger equation). The presented discrete transparent boundary conditions yield an unconditionally stable numerical scheme and are completely reflection–free at the boundary.

Comparisons of numerical methods with respect to convectively dominated problems

Abstract: A series of numerical schemes: first-order upstream, Lax-Friedrichs; second-order upstream, central difference, Lax-Wendroff, Beam–Warming, Fromm; third-order QUICK, QUICKEST and high resolution flux-corrected transport and total variation diminishing (TVD) methods are compared for one-dimensional convection-diffusion problems. Numerical results show that the modified TVD Lax-Friedrichs method is the most competent method for convectively dominated problems with a steep spatial gradient of the variables.

'Generalized Finite Differences' in Computational Electromagnetics

Abstract: The geometrical approach to Maxwell's equations promotes a way to discretize them that can be dubbed Generalized Finite Differences, which has been realized independently in several computing codes. The main features of this method are the use of two grids in duality, the metric-free formulation of the main equations (Ampere and Faraday), and the concentration of metric information in the discrete representation of the Hodge operator. The specific role that finite elements have to play in such an approach is emphasized, and a rationale for Whitney forms is proposed, showing why they are the finite elements of choice.

A finite volume approach for contingent claims valuation

Abstract: This paper presents a finite volume approach for solving two-dimensional contingent claims valuation problems. The contingent claims PDEs are in non-divergence form. The finite volume method is more flexible than finite difference schemes which are often described in the finance literature and frequently used in practice. Moreover, the finite volume method naturally handles cases where the underlying partial differential equation becomes convection dominated or degenerate. A compact method is developed which uses a high-order flux limiter for the convection terms. This paper will demonstrate how a variety of two-dimensional valuation problems can all be solved using the same approach. The generality of the approach is in part due to the fact that changes caused by different model specifications are localized. Constraints on the solution are treated in a uniform manner using a penalty method. A variety of illustrative example computations are presented.

Three-Dimensional Model of Navier-Stokes Equations for Water Waves

Abstract: A three-dimensional numerical model based on the complete Navier-Stokes equations is developed and presented in this paper. The model can be used for the problem of propagation of fully nonlinear water waves. The Navier-Stokes equations are first transformed from an irregular calculation domain to a regular one using sigma coordinates. The projection method is used to separate advection and diffusion terms from the pressure terms in Navier-Stokes equations. MacCormack's explicit scheme is used for the advection and diffusion terms, and it has second-order accuracy in both space and time. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. The resulting hydrodynamic pressure equation is solved by a multigrid method. A staggered mesh and central spatial finite-difference scheme are used. The model is tested against the experimental data of Luth et al., and the comparison shows that higher harmonics can be modeled well. Comparison of the model solutions with the elliptic shoal case confirms that the present model works well for wave refraction and diffraction with strong wave focusing.

A higher-order scheme for quasilinear boundary value problems with two small parameters

Abstract: A class of singularly perturbed quasilinear boundary value problems with two small parameters is solved numerically by finite differences on a Shishkin-type mesh. The discretization combines a four-point third-order scheme inside the boundary layers with the standard central scheme outside the layers. This results in an almost third-order accuracy which is uniform with respect to the perturbation parameters. The paper also shows that the Shishkin meshes are more suitable for higher-order schemes than the Bakhvalov meshes, since complicated nonequidistant schemes can be avoided.